GIFT   OF 


SEMICENTENNIAL  PUBLICATIONS 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 


1868-1918 


THE  THEORY  OF 
THE  RELATIVITY  OF  MOTION 


BY 

RICHARD  C.  TOLMAN 


UNIVERSITY  OF  CALIFORNIA  PRESS 
BERKELEY 

1917 


354688 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER.  PA 


TO 

H.  E. 


THE  THEORY  OF  THE  RELATIVITY  OF  MOTION. 

BY 

RICHARD  C.  TOLMAN,  PH.D. 

TABLE  OF  CONTENTS. 

PREFACE 1 

CHAPTER  I.    Historical  Development  of  Ideas  as  to  the  Nature  of  Space  and 

Time 5 

Part  I.    The  Space  and  Time  of  Galileo  and  Newton 5 

Newtonian  Time 6 

Newtonian  Space 7 

The  Galileo  Transformation  Equations 9 

Part  II.    The  Space  and  Time  of  the  Ether  Theory 10 

Rise  of  the  Ether  Theory 10 

Idea  of  a  Stationary  Ether 12 

Ether  in  the  Neighborhood  of  Moving  Bodies 12 

Ether  Entrained  in  Dielectrics 13 

The  Lorentz  Theory  of  a  Stationary  Ether 13 

Part  III.    Rise  of  the  Einstein  Theory  of  Relativity 17 

The  Michelson-Morley  Experiment 17 

The  Postulates  of  Einstein 18 

CHAPTER  II.    The  Two  Postulates  01  the  Einstein  Theory  of  Relativity 20 

The  First  Postulate  of  Relativity 20 

The  Second  Postulate  of  the  Einstein  Theory  of  Relativity 21 

Suggested  Alternative  to  the  Postulate  of  the  Independence  of  the 

Velocity  of  Light  and  the  Velocity  of  the  Source 23 

Evidence  against  Emission  Theories  of  Light 24 

Different  Forms  of  Emission  Theory 25 

Further  Postulates  of  the  Theory  of  Relativity 27 

CHAPTER  III.    Some  Elementary  Deductions 28 

Measurements  of  Tune  in  a  Moving  System 28 

Measurements  of  Length  in  a  Moving  System 30 

The  Setting  of  Clocks  in  a  Moving  System 33 

The  Composition  of  Velocities 35 

The  Mass  of  a  Moving  Body 37 

The  Relation  between  Mass  and  Energy 39 

CHAPTER  IV.    The  Einstein  Transformation  Equations  for  Space  and  Time  .  .  42 

The  Lorentz  Transformation .' 42 

Deduction  of  the  Fundamental  Transformation  Equations 43 

The  Three  Conditions  to  be  Fulfilled 44 

The  Transformation  Equations 45 

Further  Transformation  Equations 47 

Transformation  Equations  for  Velocity 47 

Transformation  Equations  for  the  Function  —    47 


vi  Table  of  Contents. 

j 

Transformation  Equations  for  Acceleration 48 

CHAPTER  V.     Kinematical  Applications 49 

The  Kinematical  Shape  of  a  Rigid  Body 49 

The  Kinematical  Rate  of  a  Clock 50 

The  Idea  of  Simultaneity 51 

The  Composition  of  Velocities 52 

The  Case  of  Parallel  Velocities 52 

Composition  of  Velocities  in  General 53 

Velocities  Greater  than  that  of  Light 54 

Applications  to  Optical  Problems 56 

The  Doppler  Effect 57 

The  Aberration  of  Light 59 

Velocity  of  Light  in  Moving  Media 60 

Group  Velocity 61 

CHAPTER  VI.    The  Dynamics  of  a  Particle 62 

The  Laws  of  Motion 62 

Difference  between  Newtonian  and  Relativity  Mechanics 62 

The  Mass  of  a  Moving  Particle 63 

Transverse  Collision 63 

Mass  the  Same  in  all  Directions 66 

Longitudinal  Collision 67 

Collision  of  any  Type 68 

Transformation  Equations  for  Mass 72 

The  Force  Acting  on  a  Moving  Particle 73 

Transformation  Equations  for  Force 73 

The  Relation  between  Force  and  Acceleration 74 

Transverse  and  Longitudinal  Acceleration 76 

The  Force  Exerted  by  a  Moving  Charge 77 

The  Field  around  a  Moving  Charge 79 

Application  to  a  Specific  Problem 80 

Work 81 

Kinetic  Energy 81 

Potential  Energy 82 

The  Relation  between  Mass  and  Energy 83 

Application  to  a  Specific  Problem 85 

CHAPTER  VII.    The  Dynamics  of  a  System  of  Particles 88 

On  the  Nature  of  a  System  of  Particles 88 

The  Conservation  of  Momentum 89 

The  Equation  of  Angular  Momentum 90 

The  Function  T 92 

The  Modified  Lagrangian  Function 93 

The  Principle  of  Least  Action 93 

Lagrange's  Equations 95 

Equations  of  Motion  in  the  Hamiltonian  Form 96 

Value  of  the  Function  T' 97 

The  Principle  of  the  Conservation  of  Energy 99 

On  the  Location  of  Energy  in  Space 100 


Table  of  Contents.  vii 

CHAPTER  VIII.     The  Chaotic  Motion  of  a  System  of  Particles 102 

The  Equations  of  Motion 102 

Representation  in  Generalized  Space 103 

Liouville's  Theorem 103 

A  System  of  Particles 104 

Probability  of  a  Given  Statistical  State 105 

Equilibrium  Relations 106 

The  Energy  as  a  Function  of  the  Momentum 108 

The  Distribution  Law 109 

Polar  Coordinates 110 

The  Law  of  Equipartition 110 

Criterion  for  Equality  of  Temperature 112 

Pressure  Exerted  by  a  System  of  Particles 113 

The  Relativity  Expression  for  Temperature 114 

The  Partition  of  Energy 117 

Partition  of  Energy  for  Zero  Mass 117 

Approximate  Partition  for  Particles  of  any  Mass 118 

CHAPTER  IX.     The  Principle  of  Relativity  and  the  Principle  of  Least  Action .  121 

The  Principle  of  Least  Action 121 

The  Equations  of  Motion  in  the  Lagrangian  Form 122 

Introduction  of  the  Principle  of  Relativity 124 

Relation  between  f  W'dt'  and  f  Wdt 124 

Relation  between  H'  and  H 127 

CHAPTER  X.  The  Dynamics  of  Elastic  Bodies 130 

On  the  Impossibility  of  Absolutely  Rigid  Bodies 130 

Part  I.  Stress  and  Strain 130 

Definition  of  Strain 130 

Definition  of  Stress 132 

Transformation  Equations  for  Strain 133 

Variation  in  the  Strain 134 

Part  II.  Introduction  of  the  Principle  of  Least  Action 137 

The  Kinetic  Potential  for  an  Elastic  Body 137 

Lagrange's  Equations 138 

Transformation  Equations  for  Stress '. . .  139 

Value  of  E° 139 

The  Equations  of  Motion  in  the  Lagrangian  Form 140 

Density  of  Momentum 142 

Density  of  Energy 142 

Summary  of  Results  from  the  Principle  of  Least  Action 142 

Part  III.  Some  Mathematical  Relations 143 

The  Unsymmetrical  Stress  Tensor  t 143 

The  Symmetrical  Tensor  p 145 

Relation  between  div  t  and  t^ 146 

The  Equations  of  Motion  in  the  Eulerian  Form 147 

Part  IV.  Applications  of  the  Results 148 

Relation  between  Energy  and  Momentum 148 

The  Conservation  of  Momentum .  .  149 


viii  Table  of  Contents. 

The  Conservation  of  Angular  Momentum 150 

Relation  between  Angular  Momentum  and  the  Unsymmetrical 

Stress  Tensor 151 

The  Right-Angled  Lever 152 

Isolated  Systems  in  a  Steady  State 154 

The  Dynamics  of  a  Particle 154 

Conclusion 154 

CHAPTER  XI.     The  Dynamics  of  a  Thermodynamic  System 156 

The  Generalized  Coordinates  and  Forces 156 

Transformation  Equation  for  Volume 156 

Transformation  Equation  for  Entropy 157 

Introduction   of  the   Principle   of   Least  Action.     The   Kinetic 

Potential 157 

The  Lagrangian  Equations 158 

Transformation  Equation  for  Pressure 159 

Transformation  Equation  for  Temperature 159 

The  Equations  of  Motion  for  Quasistationary  Adiabatic  Accelera- 
tion     160 

The  Energy  of  a  Moving  Thermodynamic  System 161 

The  Momentum  of  a  Moving  Thermodynamic  System 161 

The  Dynamics  of  a  Hohlraum 162 

CHAPTER  XII.    Electromagnetic  Theory 164 

The  Form  of  the  Kinetic  Potential 164 

The  Principle  of  Least  Action 165 

The  Partial  Integrations 165 

Derivation  of  the  Fundamental  Equations  of  Electromagnetic 

Theory 166 

The  Transformation  Equations  for  e,  h  and  p 168 

The  Invariance  of  Electric  Charge 170 

The  Relativity  of  Magnetic  and  Electric  Fields 171 

Nature  of  Electromotive  Force 172 

Derivation  of  the  Fifth  Fundamental  Equation 172 

Difference  between  the  Ether  and  the  Relativity  Theories  of  Electro- 
magnetics     173 

Applications  to  Electromagnetic  Theory 176 

The  Electric  and  Magnetic  Fields  around  a  Moving  Charge 176 

The  Energy  of  a  Moving  Electromagnetic  System 178 

Relation  between  Mass  and  Energy 180 

The  Theory  of  Moving  Dielectrics 181 

Relation    between   Field    Equations    for    Material    Media    and 

Electron  Theory 182 

Transformation  Equations  for  Moving  Media 183 

Theory  of  the  Wilson  Experiment 186 

CHAPTER  XIII.     Four-Dimensional  Analysis 188 

Idea  of  a  Time  Axis 188 

Non-Euclidean  Character  of  the  Space 189 


Table  of  Contents.  ix 

Part  I.     Vector  Analysis  of  the  Non-Euclidean  Four-Dimensional  Mani- 
fold    191 

Space,  Time  and  Singular  Vectors 192 

Invariance  of  z2  +  t/2  +  z2  -  c2*2 192 

Inner  Product  of  One-Vectors 193 

Non-Euclidean  Angle 194 

Kinematical  Interpretation  of  Angle  in  Terms  of  Velocity 194 

Vectors  of  Higher  Dimensions 195 

Outer  Products 195 

Inner  Product  of  Vectors  in  General 198 

The  Complement  of  a  Vector 198 

The  Vector  Operator,  O  or  Quad 199 

Tensors 200 

The  Rotation  of  Axes 201 

Interpretation  of  the  Lorentz  Transformation  as  a  Rotation  of 

Axes 206 

Graphical  Representation 208 

Part  II.     Applications  of  the  Four-Dimensional  Analysis 211 

Kinematics 211 

Extended  Position 211 

Extended  Velocity 212 

Extended  Acceleration 213 

The  Velocity  of  Light 214 

The  Dynamics  of  a  Particle 214 

Extended  Momentum 214 

The  Conservation  Laws 215 

The  Dynamics  of  an  Elastic  Body 216 

The  Tensor  of  Extended  Stress 216 

The  Equation  of  Motion 216 

Electromagnetics * 217 

Extended  Current 218 

The  Electromagnetic  Vector  M 217 

The  Field  Equations • 217 

The  Conservation  of  Electricity 218 

The  Product  M-q 218 

The  Extended  Tensor  of  Electromagnetic  Stress 219 

Combined  Electrical  and  Mechanical  Systems 221 

Appendix  I.     Symbols  for  Quantities 222 

Scalar  Quantities 222 

Vector  Quantities 223 

Appendix  II.     Vector  Notation 224 

Three  Dimensional  Space 224 

Non-Euclidean  Four  Dimensional  Space 225 


PREFACE. 

Thirty  or  forty  years  ago,  in  the  field  of  physical  science,  there 
was  a  widespread  feeling  that  the  days  of  adventurous  discovery  had 
passed  forever,  and  the  conservative  physicist  was  only  too  happy  to 
devote  his  life  to  the  measurement  to  the  sixth  decimal  place  of 
quantities  whose  significance  for  physical  theory  was  already  an  old 
story.  The  passage  of  time,  however,  has  completely  upset  such 
bourgeois  ideas  as  to  the  state  of  physical  science,  through  the  dis- 
covery of  some  most  extraordinary  experimental  facts  and  the  develop- 
ment of  very  fundamental  theories  for  their  explanation. 

On  the  experimental  side,  the  intervening  years  have  seen  the 
discovery  of  radioactivity,  the  exhaustive  study  of  the  conduction  of 
electricity  through  gases,  the  accompanying  discoveries  of  cathode, 
canal  and  X-rays,  the  isolation  of  the  electron,  the  study  of  the 
distribution  of  energy  in  the  hohlraum,  and  the  final  failure  of  all 
attempts  to  detect  the  earth's  motion  through  the  supposititious 
ether.  During  this  same  time,  the  theoretical  physicist  has  been 
working  hand  in  hand  with  the  experimenter  endeavoring  to  correlate 
the  facts  already  discovered  and  to  point  the  way  to  further  research. 
The  theoretical  achievements,  which  have  been  found  particularly 
helpful  in  performing  these  functions  of  explanation  and  prediction, 
have  been  the  development  of  the  modern  theory  of  electrons,  the 
application  of  thermodynamic  and  statistical  reasoning  to  the  phe- 
nomena of  radiation,  and  the  development  of  Einstein's  brilliant 
theory  of  the  relativity  of  motion. 

It  has  been  the  endeavor  of  the  following  book  to  present  an 
introduction  to  this  theory  of  relativity,  which  in  the  decade  since 
the  publication  of  Einstein's  first  paper  in  1905  (Annalen  der  Physik) 
has  become  a  necessary  part  of  the  theoretical  equipment  of  every 
physicist.  Even  if  we  regard  the  Einstein  theory  of  relativity  merely 
as  a  convenient  tool  for  the  prediction  of  electromagnetic  and  optical 
phenomena,  its  importance  to  the  physicist  is  very  great,  not  only 
because  its  introduction  greatly  simplifies  the  deduction  of  many 

2  1 


2     '  .,-.•'    Preface. 

theorems  which  were  already  familiar  in  the  older  theories  based  on  a 
stationary  ether,  but  also  because  it  leads  simply  and  directly  to  cor- 
rect conclusions  in  the  case  of  such  experiments  as  those  of  Michelson 
and  Morley,  Trouton  and  Noble,  and  Kaufman  and  Bucherer,  which 
can  be  made  to  agree  with  the  idea  of  a  stationary  ether  only  by  the 
introduction  of  complicated  and  ad  hoc  assumptions.  Regarded  from 
a  more  philosophical  point  of  view,  an  acceptance  of  the  Einstein 
theory  of  relativity  shows  us  the  advisability  of  completely  remodelling 
some  of  our  most  fundamental  ideas.  In  particular  we  shall  now 
do  well  to  change  our  concepts  of  space  and  time  in  such  a  way  as 
to  give  up  the  old  idea  of  their  complete  independence,  a  notion 
which  we  have  received  as  the  inheritance  of  a  long  ancestral  experience 
with  bodies  moving  with  slow  velocities,  but  which  no  longer  proves 
pragmatic  when  we  deal  with  velocities  approaching  that  of  light. 

The  method  of  treatment  adopted  in  the  following  chapters  is 
to  a  considerable  extent  original,  partly  appearing  here  for  the  first 
time  and  partly  already  published  elsewhere.*  Chapter  III  follows 
a  method  which  was  first  developed  by  Lewis  and  Tolman,f  and  the 
last  chapter  a  method  developed  by  Wilson  and  Lewis.  $  The  writer 
must  also  express  his  special  obligations  to  the  works  of  Einstein, 
Planck,  Poincare,  Laue,  Ishiwara  and  Laub. 

It  is  hoped  that  the  mode  of  presentation  is  one  that  will  be  found 
well  adapted  not  only  to  introduce  the  study  of  relativity  theory  to 
those  previously  unfamiliar  with  the  subject  but  also  to  provide  the 
necessary  methodological  equipment  for  those  who  wish  to  pursue 
the  theory  into  its  more  complicated  applications. 

After  presenting,  in  the  first  chapter,  a  brief  outline  of  the  historical 
development  of  ideas  as  to  the  nature  of  the  space  and  time  of  science, 
we  consider,  in  Chapter  II,  the  two  main  postulates  upon  which  the 
theory  of  relativity  rests  and  discuss  the  direct  experimental  evidence 
for  their  truth.  The  third  chapter  then  presents  an  elementary  and 

*  Philosophical  Magazine,  vol.  18,  p.  510  (1909);  Physical  Review,  vol.  31,  p.  26 
(1910);  Phil.  Mag.,  vol.  21,  p.  296  (1911);  ibid.,  vol.  22,  p.  458  (1911);  ibid.,  vol.  23, 
p.  375  (1912);  Phys.  Rev.,  vol.  35,  p.  136  (1912);  Phil.  Mag.,  vol.  25,  p.  150  (1913); 
ibid.,  vol.  28,  p.  572  (1914);  ibid.,  vol.  28,  p.  583  (1914). 

t  Phil.  Mag.,  vol.  18,  p.  510  (1909). 

t  Proceedings  of  the  American  Academy  of  Arts  and  Sciences,  vol.  48,  p.  389 
(1912). 


Preface.  3 

non-mathematical  deduction  of  a  number  of  the  most  important 
consequences  of  the  postulates  of  relativity,  and  it  is  hoped  that  this 
chapter  will  prove  especially  valuable  to  readers  without  unusual 
mathematical  equipment,  since  they  will  there  be  able  to  obtain  a 
real  grasp  of  such  important  new  ideas  as  the  change  of  mass  with 
velocity,  the  non-additivity  of  velocities,  and  the  relation  of  mass 
and  energy,  without  encountering  any  mathematics  beyond  the 
elements  of  analysis  and  geometry. 

In  Chapter  IV  we  commence  the  more  analytical  treatment  of 
the  theory  of  relativity  by  obtaining  from  the  two  postulates  of 
relativity  Einstein's  transformation  equations  for  space  and  time  as 
well  as  transformation  equations  for  velocities,  accelerations,  and 
for  an  important  function  of  the  velocity.  Chapter  V  presents 
various  kinematical  applications  of  the  theory  of  relativity  following 
quite  closely  Einstein's  original  method  of  development.  In  par- 
ticular we  may  call  attention  to  the  ease  with  which  we  may  handle 
the  optics  of  moving  media  by  the  methods  of  the  theory  of  relativity 
as  compared  with  the  difficulty  of  treatment  on  the  basis  of  the  ether 
theory. 

In  Chapters  VI,  VII  and  VIII  we  develop  and  apply  a  theory  of 
the  dynamics  of  a  particle  which  is  based  on  the  Einstein  trans- 
formation equations  for  space  and  time,  Newton's  three  laws  of  motion, 
and  the  principle  of  the  conservation  of  mass. 

We  then  examine,  in  Chapter  IX,  the  relation  between  the  theory 
of  relativity  and  the  principle  of  least  action,  and  find  it  possible  to 
introduce  the  requirements  of  relativity  theory  at  the  very  start  into 
this  basic  principle  for  physical  science.  We  point  out  that  we 
might  indeed  have  used  this  adapted  form  of  the  principle  of  least 
action,  for'  developing  the  dynamics  of  a  particle,  and  then  proceed 
in  Chapters  X,  XI  and  XII  to  develop  the  dynamics  of  an  elastic 
body,  the  dynamics  of  a  thermodynamic  system,  and  the  dynamics 
of  an  electromagnetic  system,  all  on  the  basis  of  our  adapted  form 
of  the  principle  of  least  action. 

Finally,  in  Chapter  XIII,  we  consider  a  four-dimensional  method 
of  expressing  and  treating  the  results  of  relativity  theory.  This 
chapter  contains,  in  Part  I,  an  epitome  of  some  of  the  more  important 
methods  in  four-dimensional  vector  analysis  and  it  is  hoped  that  it 


4  Preface. 

can  also  be  used  in  connection  with  the  earlier  parts  of  the  book  as  a 
convenient  reference  for  those  who  are  not  familiar  with  ordinary 
three-dimensional  vector  analysis. 

In  the  present  book,  the  writer  has  confined  his  considerations  to 
cases  in  which  there  is  a  uniform  relative  velocity  between  systems  of 
coordinates.  In  the  future  it  may  be  possible  greatly  to  extend  the 
applications  of  the  theory  of  relativity  by  considering  accelerated 
systems  of  coordinates,  and  in  this  connection  Einstein's  latest  work 
on  the  relation  between  gravity  and  acceleration  is  of  great  interest. 
It  does  not  seem  wise,  however,  at  the  present  time  to  include  such 
considerations  in  a  book  which  intends  to  present  a  survey  of  accepted 
theory. 

The  author  will  feel  amply  repaid  for  the  work  involved  in  the 
preparation  of  the  book  if,  through  his  efforts,  some  of  the  younger 
American  physicists  can  be  helped  to  obtain  a  real  knowledge  of  the 
important  work  of  Einstein.  He  is  also  glad  to  have  this  opportunity 
to  add  his  testimony  to  the  growing  conviction  that  the  conceptual 
space  and  time  of  science  are  not  God-given  and  unalterable,  but  are 
rather  in  the  nature  of  human  constructs  devised  for  use  in  the  de- 
scription and  correlation  of  scientific  phenomena,  and  that  these 
spatial  and  temporal  concepts  should  be  altered  whenever  the  discovery 
of  new  facts  makes  such  a  change  pragmatic. 

The  writer  wishes  to  express  his  indebtedness  to  Mr.  William  H. 
Williams  for  assisting  in  the  preparation  of  Chapter  I. 


CHAPTER  I. 

HISTORICAL   DEVELOPMENT    OF    IDEAS    AS    TO    THE    NATURE    OF 

SPACE  AND  TIME. 

1.  Since  the  year  1905,  which  marked  the  publication  of  Einstein's 
momentous  article  on  the  theory  of  relativity,  the  development  of 
scientific  thought  has  led  to  a  complete  revolution  in  accepted  ideas 
as  to  the  nature  of  space  and  time,  and  this  revolution  has  in  turn 
profoundly  modified  those  dependent  sciences,  in  particular  mechanics 
and  electromagnetics,   which  make  use  of  these  two  fundamental 
concepts  in  their  considerations. 

In  the  following  pages  it  will  be  our  endeavor  to  present  a  de- 
scription of  these  new  notions  as  to  the  nature  of  space  and  time,* 
and  to  give  a  partial  account  of  the  consequent  modifications  which 
have  been  introduced  into  various  fields  of  science.  Before  pro- 
ceeding to  this  task,  however,  we  may  well  review  those  older  ideas 
as  to  space  and  time  which  until  now  appeared  quite  sufficient  for 
the  correlation  of  scientific  phenomena.  We  shall  first  consider  the 
space  and  time  of  Galileo  and  Newton  which  were  employed  in  the 
development  of  the  classical  mechanics,  and  then  the  space  and  time 
of  the  ether  theory  of  light. 

PART  I.     THE  SPACE  AND  TIME  OF  GALILEO  AND  NEWTON. 

2.  The  publication  in  1687  of  Newton's  Principia  laid  down  so 
satisfactory  a  foundation  for  further  dynamical  considerations,  that 
it  seemed  as  though  the  ideas  of  Galileo  and  Newton  as  to  the  nature 
of  space  and  time,  which  were  there  employed,  would  certainly  remain 
forever  suitable  for  the  interpretation  of  natural  phenomena.     And 
indeed  upon  this  basis  has  been  built  the  whole  structure  of  classical 
mechanics  which,  until  our  recent  familiarity  with  very  high  velocities, 
has  been  found  completely  satisfactory  for  an  extremely' large  number 
of  very  diverse  dynamical  considerations. 

*  Throughout  this  work  by  "space"  and  "time"  we  shall  mean  the  conceptual 
space  and  time  of  science. 

5 


6  Chapter  One. 

An  examination  of  the  fundamental  laws  of  mechanics  will  show 
how  the  concepts  of  space  and  time  entered  into  the  Newtonian 
system  of  mechanics.  Newton's  laws  of  motion,  from  which  the 
whole  of  the  classical  mechanics  could  be  derived,  can  best  be  stated 
with  the  help  of  the  equation 

F=|(wm).  (1) 

This  equation  defines  the  force  F  acting  on  a  particle  as  equal  to  the 
rate  of  change  in  its  momentum  (i.  e.,  the  product  of  its  mass  m  and 
its  velocity  u),  and  the  whole  of  Newton's  laws  of  motion  may  be 
summed  up  in  the  statement  that  in  the  case  of  two  interacting  par- 
ticles the  forces  which  they  mutually  exert  on  each  other  are  equal  in 
magnitude  and  opposite  in  direction. 

Since  in  Newtonian  mechanics  the  mass  of  a  particle  is  assumed 
constant,  equation  (1)  may  be  more  conveniently  written 


d\i          d  fdr 

=  m  -j-  =  m  -r  1  TT 

dt  dtdt 


\ 

I  , 
J 


or 


and  this  definition  of  force,  together  with  the  above-stated  principle 
of  the  equality  of  action  and  reaction,  forms  the  starting-point  for 
the  whole  of  classical  mechanics. 

The  necessary  dependence  of  this  mechanics  upon  the  concepts 
of  space  and  time  becomes  quite  evident  on  an  examination  of  this 
fundamental  equation  (2) ,  in  which  the  expression  for  the  force  acting 
on  a  particle  is  seen  to  contain  both  the  variables  x,  y,  and  z,  which 
specify  the  position  of  the  particle  in  space,  and  the  variable  t,  which 
specifies  the  time. 

3.  Newtonian  Time.     To  attempt  a  definite  statement  as  to  the 


Historical  Development.  7 

meaning  of  so  fundamental  and  underlying  a  notion  as  that  of  time 
is  a  task  from  which  even  philosophy  may  shrink.  In  a  general 
way,  conceptual  time  may  be  thought  of  as  a  one-dimensional,  uni- 
directional, one-valued  continuum.  This  continuum  is  a  sort  of  frame- 
work in  which  the  instants  at  which  actual  occurrences  take  place 
find  an  ordered  position.  Distances  from  point  to  point  in  the 
continuum,  that  is  intervals  of  time,  are  measured  by  the  periods  of 
certain  continually  recurring  cyclic  processes  such  as  the  daily  rota- 
tion of  the  earth.  A  unidirectional  nature  is  imposed  upon  the  time 
continuum  among  other  things  by  an  acceptance  of  the  second  law 
of  thermodynamics,  which  requires  that  actual  progression  in  time 
shall  be  accompanied  by  an  increase  in  the  entropy  of  the  material 
world,  and  this  same  law  requires  that  the  continuum  shall  be  one- 
valued  since  it  excludes  the  possibility  that  time  ever  returns  upon 
itself,  either  to  commence  a  new  cycle  or  to  intersect  its  former  path 
even  at  a  single  point. 

In  addition  to  these  characteristics  of  the  time  continuum,  which 
have  been  in  no  way  modified  by  the  theory  of  relativity,  the  New- 
tonian mechanics  always  assumed  a  complete  independence  of  time  and 
the  three-dimensional  space  continuum  which  exists  along  with  it. 
In  dynamical  equations  time  entered  as  an  entirely  independent  vari- 
able in  no  way  connected  with  the  variables  whose  specification 
determines  position  in  space.  In  the  following  pages,  however,  we 
shall  find  that  the  theory  of  relativity  requires  a  very  definite  inter- 
relation between  time  and  space,  and  in  the  Einstein  transformation 
equations  we  shall  see  the  exact  way  in  which  measurements  of  time 
depend  upon  the  choice  of  a  set  of  variables  for  measuring  position 
in  space. 

4.  Newtonian  Space.  An  exact  description  of  the  concept  of 
space  is  perhaps  just  as  difficult  as  a  description  of  the  concept  of  time. 
In  a  general  way  we  think  of  space  as  a  three-dimensional,  homo- 
geneous, isotropic  continuum,  and  these  ideas  are  common  to  the 
conceptual  spaces  of  Newton,  Einstein,  and  the  ether  theory  of  light. 
The  space  of  Newton,  however,  differs  on  the  one  hand  from  that  of 
Einstein  because  of  a  tacit  assumption  of  the  complete  independence 
of  space  and  time  measurements;  and  differs  on  the  other  hand  from 
that  of  the  ether  theory  of  light  by  the  fact  that  "  free  "  space  was 


8  Chapter  One. 

assumed  completely  empty  instead  of  filled  with  an  all-pervading 
quasi-material  medium — the  ether.  A  more  definite  idea  of  the  par- 
ticularly important  characteristics  of  the  Newtonian  concept  of  space 
may  be  obtained  by  considering  somewhat  in  detail  the  actual  methods 
of  space  measurement. 

Positions  in  space  are  in  general  measured  with  respect  to  some 
arbitrarily  fixed  system  of  reference  which  must  be  threefold  in 
character  corresponding  to  the  three  dimensions  of  space.  In  par- 
ticular we  may  make  use  of  a  set  of  Cartesian  axes  and  determine, 
for  example,  the  position  of  a  particle  by  specifying  its  three  Cartesian 
coordinates  x,  y  and  z. 

In  Newtonian  mechanics  the  particular  set  of  axes  chosen  for 
specifying  position  in  space  has  in  general  been  determined  in  the 
first  instance  by  considerations  of  convenience.  For  example,  it  is 
found  by  experience  thai^  if  we  take  as  a  reference  system  lines  drawn 
upon  the  surface  of  the  earth,  the  equations  of  motion  based  on  New- 
tor?s  laws  give  us  a  simple  description  of  nearly  all  dynamical  phe- 
nomena which  are  merely  terrestrial.  When,  however,  we  try  to 
interpret  with  these  same  axes  the  motion  of  the  heavenly  bodies,  we 
meet  difficulties,  and  the  problem. is  simplified,  so  far  as  planetary 
motions  are  concerned,  by  taking  a  new  reference  system  determined 
by  the  sun  and  the  fixed  stars.  But  this  system,  in  its  turn,  becomes 
somewhat  unsatisfactory  when  we  take  account  of  the  observed 
motions  of  the  stars  themselves,  and  it  is  finally  convenient  to  take  a 
reference  system  relative  to  which  the  sun  is  moving  with  a  velocity 
of  twelve  miles  per  second  in  the  direction  of  the  constellation  Hercules. 
This  system  of  axes  is  so  chosen  that  the  great  majority  of  stars  have 
on  the  average  no  motion  with  respect  to  it,  and  the  actual  motion 
of  any  particular  star  with  respect  to  these  coordinates  is  called  the 
peculiar  motion  of  the  star. 

Suppose,  now,  we  have  a  number  of  such  systems  of  axes  in  uni- 
form relative  motion;  we  are  confronted  by  the  problem  of  finding 
some  method  of  transposing  the  description  of  a  given  kinematical 
occurrence  from  the  variables  of  one  of  these  sets  of  axes  to  those  of 
another.  For  example,  if  we  have  chosen  a  system  of  axes  S  and 
have  found  an  equation  in  x,  y,  z,  and  t  which  accurately  describes  the 
motion  of  a  given  point,  what  substitutions  for  the  quantities  involved 


Historical  Development.  9 

can  be  made  so  that  the  new  equation  thereby  obtained  will  again 
correctly  describe  the  same  phenomena  when  we  measure  the  dis- 
placements of  the  point  relative  to  a  new  system  of  reference  S' 
which  is  in  uniform  motion  with  respect  to  /S?  The  assumption  of 
Galileo  and  Newton  that  "  free  "  space  is  entirely  empty,  and  the 
further  tacit  assumption  of  the  complete  independence  of  space  and 
time,  led  them  to  propose  a  very  simple  solution  of  the  problem,  and 
the  transformation  equations  which  they  used  are  generally  called 
the  Galileo  Transformation  Equations  to  distinguish  them  from  the 
Einstein  Transformation  Equations  which  we  shall  later  consider. 

5.  The  Galileo  Transformation  Equations.  Consider  two  systems 
of  right-angled  coordinates,  S  and  S',  which  are  in  relative  motion  in 
the  X  direction  with  the  velocity  F;  for  convenience  let  the  X  axes, 
OX  and  O'X',  of  the  two  systems  coincide  in  direction,  and  for  further 
simplification  let  us  take  as  our  zero  point  for  time  measurements  the 
instant  when  the  two  origins  0  and  0'  coincide.  Consider  now  a 
point  which  at  the  time  t  has  the  coordinates  x,  y  and  z  measured  in 
system  S.  Then,  according  to  the  space  and  time  considerations  of 
Galileo  and  Newton,  the  coordinates  of  the  point  with  reference  to 
system  S'  are  given  by  the  following  transformation  equations: 

x'  =  x  -  Vt,  (3) 

y'  =  y,  (4) 

*'  =  z,  (5) 

*'  =  t.  (6) 

These  equations  are  fundamental  for  Newtonian  mechanics,  and  may 
appear  to  the  casual  observer  to  be  self-evident  and  bound  up  with 
necessary  ideas  as  to  the  nature  of  space  and  time.  Nevertheless, 
the  truth  of  the  first  and  the  last  of  these  equations  is  absolutely 
dependent  on  the  unsupported  assumption  of  the  complete  inde- 
pendence of  space  and  time  measurements,  and  since  in  the  Einstein 
theory  we  shall  find  a  very  definite  relation  between  space  and  time 
measurements  we  shall  be  led  to  quite  a  different  set  of  transformation 
equations.  Relations  (3),  (4),  (5)  and  (6)  will  be  found,  however,  to 
be  the  limiting  form  which  the  correct  transformation  equations  as- 
sume when  the  velocity  between  the  systems  V  becomes  small  com- 


10  Chapter  One. 

pared  with  that  of  light.  Since  until  very  recent  times  the  human 
race  in  its  entire  past  history  has  been  familiar  only  with  velocities 
that  are  small  compared  with  that  of  light,  it  need  not  cause  surprise 
that  the  above  equations,  which  are  true  merely  at  the  limit,  should 
appear  so  self-evident. 

6.  Before  leaving  the  discussion  of  the  space  and  time  system  of 
Newton  and  Galileo  we  must  call  attention  to  an  important  charac- 
teristic which  it  has  in  common  with  the  system  of  Einstein  but 
which  is  not  a  feature  of  that  assumed  by  the  ether  theory.     If  we 
have  two  systems  of  axes  such  as  those  we  have  just  been  considering, 
we  may  with  equal  right  consider  either  one  of  them  at  rest  and  the 
other  moving  past  it.     All  we  can  say  is  that  the  two  systems  are  in 
relative  motion;  it  is  meaningless  to  speak  of  either  one  as  in  any 
sense   "  absolutely  "   at  rest.     The  equation  x'  —  x  —  Vt  which  we 
use  in  transforming  the  description  of  a  kinematical  event  from  the 
variables  of  system  S  to  those  of  system  S'  is  perfectly  symmetrical 
with  the  equation  x  =  xf  +  Vt'  which  we  should  use  for  a  trans- 
formation in  the  reverse  direction.     Of  all  possible  systems  no  par- 
ticular set  of  axes  holds  a  unique  position  among  the  others.     We 
shall  later  find  that  this  important  principle  of  the  relativity  of  motion 
is  permanently  incorporated  into  our  system  of  physical  science  as 
the  first  postulate  of  relativity.     This  principle,  common  both  to  the 
space  of  Newton  and  to  that  of  Einstein,  is  not  characteristic  of  the 
space  assumed  by  the  classical  theory  of  light.     The  space  of  this 
theory  was  supposed  to  be  filled  with  a  stationary  medium,   the 
luminiferous  ether,  and  a  system  of  axes  stationary  with  respect  to 
this  ether  would  hold  a  unique  position  among  the  other  systems 
and  be  the  one  peculiarly  adapted  for  use  as  the  ultimate  system  of 
reference  for  the  measurement  of  motions. 

We  may  now  briefly  sketch  the  rise  of  the  ether  theory  of  light  and 
point  out  the  permanent  contribution  which  it  has  made  to  physical 
science,  a  contribution  which  is  now  codified  as  the  second  postulate 
of  relativity. 

PART  II.    THE  SPACE  AND  TIME  OF  THE  ETHER  THEORY. 

7.  Rise  of  the  Ether  Theory.     Twelve  years  before  the  appearance 
of  the  Principia,  Homer,  a  Danish  astronomer,  observed  that  an 


Historical  Development.  11 

eclipse  of  one  of  the  satellites  of  Jupiter  occurred  some  ten  minutes 
later  than  the  time  predicted  for  the  event  from  the  known  period 
of  the  satellite  and  the  time  of  the  preceding  eclipse.  He  explained 
this  delay  by  the  hypothesis  that  it  took  light  twenty-two  minutes 
to  travel  across  the  earth's  orbit.  Previous  to  Romer's  discovery, 
light  was  generally  supposed  to  travel  with  infinite  velocity.  Indeed 
Galileo  had  endeavored  to  find  the  speed  of  light  by  direct  experiments 
over  distances  of  a  few  miles  and  had  failed  to  detect  any  lapse  of 
time  between  the  emission  of  a  light  flash  from  a  source  and  its  ob- 
servation by  a  distant  observer.  Romer's  hypothesis  has  been  re- 
peatedly verified  and  the  speed  of  light  measured  by  different  methods 
with  considerable  exactness.  The  mean  of  the  later  determinations 
is  2.9986  X  1010  cm.  per  second. 

8.  At  the  time  of  Romer's  discovery  there  was  much  discussion 
as  to  the  nature  of  light.  Newton's  theory  that  it  consisted  of  par- 
ticles or  corpuscles  thrown  out  by  a  luminous  body  was  attacked  by 
Hooke  and  later  by  Huygens,  who  advanced  the  view  that  it  was 
something  in  the  nature  of  wave  motions  in  a  supposed  space-filling 
medium  or  ether*  By  this  theory  Huygens  was  able  to  explain 
reflection  and  refraction  and  the  phenomena  of  color,  but  assuming 
longitudinal  vibrations  he  was  unable  to  account  for  polarization. 
Diffraction  had  not  yet  been  observed  and  Newton  contested  the 
Hooke-Huygens  theory  chiefly  on  the  grounds  that  it  was  contra- 
dicted by  the  fact  of  rectilinear  propagation  and  the  formation  of 
shadows.  The  scientific  prestige  of  Newton  was  so  great  that  the 
emission  or  corpuscular  theory  continued  to  hold  its  ground  for  a 
hundred  and  fifty  years.  Even  the  masterly  researches  of  Thomas 
Young  at  the  beginning  of  the  nineteenth  century  were  unable  to 
dislodge  the  old  theory,  and  it  was  not  until  the  French  physicist, 
Fresnel,  about  1815,  was  independently  led  to  an  undulatory  theory 
and  added  to  Young's  arguments  the  weight  of  his  more  searching 
mathematical  analysis,  that  the  balance  began  to  turn.  From  this 
time  on  the  wave  theory  grew  in  power  and  for  a  period  of  eighty 
years  was  not  seriously  questioned.  This  theory  has  for  its  essential 
postulate  the  existence  of  an  all-pervading  medium,  the  ether,  in 
which  wave  disturbances  can  be  set  up  and  propagated.  And  the 
physical  properties  of  this  medium  became  an  enticing  field  of  inquiry 
and  speculation. 


12  Chapter  One. 

9.  Idea  of  a  Stationary  Ether.     Of  all  the  various  properties  with 
which  the  physicist  found  it  necessary  to  endow  the  ether,  for  us  the 
most  important  is  the  fact  that  it  must  apparently  remain  stationary, 
unaffected  by  the  motion  of  matter  through  it.     This  conclusion  was 
finally  reached  through  several  lines  of  investigation.     We  may  first 
consider  whether  the  ether  would  be  dragged  along  by  the  motion  of 
nearby  masses  of  matter,  and,  second,  whether  the  ether  enclosed  in  a 
moving  medium  such  as  water  or  glass  would  partake  in  the  latter's 
motion. 

10.  Ether  in  the   Neighborhood  of  Moving  Bodies.     About  the 
year   1725  the  astronomer  Bradley,   in  his  efforts  to  measure  the 
parallax  of  certain  fixed  stars,  discovered  that  the  apparent  position 
of  a  star  continually  changes  in  such  a  way  as  to  trace  annually  a 
small  ellipse  in  the  sky,  the  apparent  position  always  lying  in  the 
plane  determined  by  the  line  from  the  earth  to  the  center  of  the 
ellipse  and  by  the  direction  of  the  earth's  motion.     On  the  corpuscular 
theory  of  light  this  admits  of  ready  explanation  as  Bradley  himself 
discovered,  since  we  should  expect  the  earth's  motion  to  produce  an 
apparent  change  in  the  direction  of  the  oncoming  light,  in  just  the 
same  way  that  the  motion  of  a  railway  train  makes  the  falling  drops 
of  rain  take  a  slanting  path  across  the  window  pane.     If  c  be  the 
velocity  of  a  light  particle  and  v  the  earth's  velocity,  the  apparent  or 
relative  velocity  would  be  c  —  v  and  the  tangent  of  the  angle  of 

aberration  would  be  - . 
c 

Upon  the  wave  theory,  it  is  obvious  that  we  should  also  expect  a 
similar  aberration  of  light,  provided  only  that  the  ether  shall  be 
quite  stationary  and  unaffected  by  the  motion  of  the  earth  through  it, 
and  this  is  one  of  the  important  reasons  that  most  ether  theories  have 
assumed  a  stationary  ether  unaffected  by  the  motion  of  neighboring 
matter.  * 

In  more  recent  years  further  experimental  evidence  for  assuming 
that  the  ether  is  not  dragged  along  by  the  neighboring  motion  of 
large  masses  of  matter  was  found  by  Sir  Oliver  Lodge.  His  final 
experiments  were  performed  with  a  large  rotating  spheroid  of  iron 

*  The  most  notable  exception  is  the  theory  of  Stokes,  which  did  assume  that 
the  ether  moved  along  with  the  earth  and  then  tried  to  account  for  aberration  with 
the  help  of  a  velocity  potential,  but  this  led  to  difficulties,  as  was  shown  by  Lorentz. 


Historical  Development.  13 

with  a  narrow  groove  around  its  equator,  which  was  made  the  path 
for  two  rays  of  light,  one  travelling  in  the  direction  of  rotation  and 
the  other  in  the  opposite  direction.  Since  by  interference  methods 
no  difference  could  be  detected  in  the  velocities  of  the  two  rays,  here 
also  the  conclusion  was  reached  that  the  ether  was  not  appreciably 
dragged  along  by  the  rotating  metal. 

11.  Ether  Entrained  in  Dielectrics.     With  regard  to  the  action  of 
a  moving  medium  on  the  ether  which  might  be  entrained  within  it, 
experimental  evidence  and  theoretical  consideration  here  too  finally 
led  to  the  supposition  that  the  ether  itself  must  remain  perfectly 
stationary.     The  earlier  view  first  expressed  by  Fresnel,  in  a  letter 
written  to  Arago  in  1818,  was  that  the  entrained  ether  did  receive  a 
fraction  of  the  total  velocity  of  the  moving  medium.     Fresnel  gave 

to  this  fraction  the  value  —         ,  where  /*  is  the  index  of  refraction  of 

M2 

the  substance  forming  the  medium.  On  this  supposition,  Fresnel 
was  able  to  account  for  the  fact  that  Arago's  experiments  upon  the 
reflection  and  refraction  of  stellar  rays  show  no  influence  whatever 
of  the  earth's  motion,  and  for  the  fact  that  Airy  found  the  same  angle 
of  aberration  with  a  telescope  filled  with  water  as  with  air.  More- 
over, the  later  work  of  Fizeau  and  the  accurate  determinations  of 
Michelson  and  Morley  on  the  velocity  of  light  in  a  moving  stream 
of  water  did  show  that  the  speed  was  changed  by  an  amount  corre- 
sponding to  FresnePs  fraction.  The  fuller  theoretical  investigations 
of  Lorentz,  however,  did  not  lead  scientists  to  look  upon  this  increased 
velocity  of  light  in  a  moving  medium  as  an  evidence  that  the  ether 
is  pulled  along  by  the  stream  of  water,  and  we  may  now  briefly  sketch 
the  developments  which  culminated  in  the  Lorentz  theory  of  a  com- 
pletely stationary  ether. 

12.  The  Lorentz  Theory  of  a  Stationary  Ether.     The  considera- 
tions of  Lorentz  as  to  the  velocity  of  light  in  moving  media  became 
possible  only  after  it  was  evident  that  optics  itself  is  a  branch  of  the 
wider  science  of  electromagnetics,  and  it  became  possible  to  treat 
transparent  media  as  a  special  case  of  dielectrics  in  general.     In  1873, 
in  his  Treatise  on  Electricity  and  Magnetism,  Maxwell  first  advanced 
the  theory  that  electromagnetic  phenomena  also  have  their  seat  in 
the  luminiferous  ether  and  further  that  light  itself  is  merely  an  electro- 


14  Chapter  One. 

magnetic  disturbance  in  that  medium,  and  Maxwell's  theory  was 
confirmed  by  the  actual  discovery  of  electromagnetic  waves  in  1888 
by  Hertz. 

The  attack  upon  the  problem  of  the  relative  motion  of  matter  and 
ether  was  now  renewed  with  great  vigor  both  theoretically  and  experi- 
mentally from  the  electromagnetic  side.  Maxwell  in  his  treatise  had 
confined  himself  to  phenomena  in  stationary  media.  Hertz,  however, 
extended  Maxwell's  considerations  to  moving  matter  on  the  assump- 
tion that  the  entrained  ether  is  carried  bodily  along  by  it.  It  is  evi- 
dent, however,  that  in  the  field  of  optical  theory  such  an  assumption 
could  not  be  expected  to  account  for  the  Fizeau  experiment,  which 
had  already  been  explained  on  the  assumption  that  the  ether  receives 
only  a  fraction  of  the  velocity  of  the  moving  medium;  while  in  the 
field  of  electromagnetic  theory  it  was  found  that  Hertz's  assumptions 
would  lead  us  to  expect  no  production  of  a  magnetic  field  in  the 
neighborhood  of  a  rotating  electric  condenser  providing  the  plates  of 
the  condenser  and  the  dielectric  move  together  with  the  same  speed 
and  this  was  decisively  disproved  by  the  experiment  of  Eichenwald. 
The  conclusions  of  the  Hertz  theory  were  also  out  of  agreement  with 
the  important  experiments  of  H.  A.  Wilson  on  moving  dielectrics. 
It  remained  for  Lorentz  to  develop  a  general  theory  for  moving 
dielectrics  which  was  consistent  with  the  facts. 

The  theory  of  Lorentz  developed  from  that  of  Maxwell  by  the 
addition  of  the  idea  of  the  electron,  as  the  atom  of  electricity,  and  his 
treatment  is  often  called  the  "  electron  theory."  This  atomistic 
conception  of  electricity  was  foreshadowed  by  Faraday's  discovery 
of  the  quantitative  relations  between  the  amount  of  electricity  asso- 
ciated with  chemical  reactions  in  electrolytes  and  the  weight  of 
substance  involved,  a  relation  which  indicates  that  the  atoms  act  as 
carriers  of  electricity  and  that  the  quantity  of  electricity  carried  by  a 
single  particle,  whatever  its  nature,  is  always  some  small  multiple  of  a 
definite  quantum  of  electricity,  the  electron.  Since  Faraday's  time, 
the  study  of  the  phenomena  accompanying  the  conduction  of  elec- 
tricity through  gases,  the  study  of  radioactivity,  and  finally  indeed 
the  isolation  and  exact  measurement  of  these  atoms  of  electrical 
charge^  have  led  us  to  a  very  definite  knowledge  of  many  of  the 
properties  of  the  electron* 


Historical  Development.  15 

While  the  experimental  physicists  were  at  work  obtaining  this 
more  or  less  first-hand  acquaintance  with  the  electron,  the  theoretical 
physicists  and  in  particular  Lorentz  were  increasingly  successful  in 
explaining  the  electrical  and  optical  properties  of  matter  in  general 
on  the  basis  of  the  behavior  of  the  electrons  which  it  contains,  the 
properties  of  conductors  being  accounted  for  by  the  presence  of  mov- 
able electrons,  either  free  as  in  the  case  of  metals  or  combined  with 
atoms  to  form  ions  as  in  electrolytes,  while  the  electrical  and  optical 
properties  of  dielectrics  were  ascribed  to  the  presence  of  electrons 
more  or  less  bound  by  quasi-elastic  forces  to  positions  of  equilibrium* 
This  Lorentz  electron  theory  of  matter  has  been  developed  in  great 
mathematical  detail  by  Lorentz  and  has  been  substantiated  by  nu- 
merous quantitative  experiments.  Perhaps  the  greatest  significance 
of  the  Lorentz  theory  is  that  such  properties  of  matter  as  electrical 
conductivity,  magnetic  permeability  and  dielectric  inductivity,  which 
occupied  the  position  of  rather  accidental  experimental  constants  in 
Maxwell's  original  theory,  are  now  explainable  as  the  statistical  result 
of  the  behavior  of  the  individual  electrons. 

With  regard  now  to  our  original  question  as  to  the  behavior  of 
moving  optical  and  dielectric  media,  the  Lorentz  theory  was  found 
capable  of  accounting  quantitatively  for  all  known  phenomena,  in- 
cluding Airy's  experiment  on  aberration,  Arago's  experiments  on  the 
reflection  and  refraction  of  stellar  rays,  Fresnel's  coefficient  for  the 
velocity  of  light  in  moving  media,  and  the  electromagnetic  experi- 
ments upon  moving  dielectrics  made  by  Rontgen,  Eichenwald,  H.  A. 
Wilson,  and  others.  For  us  the  particular  significance  of  the  Lorentz 
method  of  explaining  these  phenomena  is  that  he  does  not  assume,  as 
did  Fresnel,  that  the  ether  is  partially  dragged  along  by  moving 
matter.  His  investigations  show  rather  that  the  ether  must  remain 
perfectly  stationary,  and  that  such  phenomena  as  the  changed  velocity 
of  light  in  moving  media  are  to  be  accounted  for  by  the  modifying 
influence  which  the  electrons  in  the  moving  matter  have  upon  the 
propagation  of  electromagnetic  disturbances,  rather  than  by  a  dragging 
along  of  the  ether  itself. 

Although  it  would  not  be  proper  in  this  place  to  present  the 
mathematical  details  of  Lorentz's  treatment  of  moving  media,  we 
may  obtain  a  clearer  idea  of  what  is  meant  in  the  Lorentz  theory  by  a 


16  Chapter  One. 

stationary  ether  if  we  look  for  a  moment  at  the  five  fundamental 
equations  upon  which  the  theory  rests.  These  familiar  equations,  of 
which  the  first  four  are  merely  Maxwell's  four  field  equations,  modified 
by  the  introduction  of  the  idea  of  the  electron,  may  be  written 

1  de         u 


i  dh 

curl  e  =  —  -  —  , 
c  dt 


div  e  =  p, 
div  h  =  0, 


[jxh]*} 


in  which  the  letters  have  their  usual  significance.  (See  Chapter  XII.) 
Now  the  whole  of  the  Lorentz  theory,  including  of  course  his  treat- 
ment of  moving  media,  is  derivable  from  these  five  equations,  and 
the  fact  that  the  idea  of  a  stationary  ether  does  lie  at  the  basis  of 
his  theory  is  most  clearly  shown  by  the  first  and  last  of  these  equa- 
tions, which  contain  the  velocity  u  with  which  the  charge  in  question 
is  moving,  and  for  Lorentz  this  velocity  is  to  be  measured  with  respect 
to  the  assumed  stationary  ether. 

We  have  devoted  this  space  to  the  Lorentz  theory,  since  his  work 
marks  the  culmination  of  the  ether  theory  of  light  and  electromag- 
netism,  and  for  us  the  particularly  significant  fact  is  that  by  this 
line  of  attack  science  was  inevitably  led  to  the  ide'a  of  an  absolutely 
immovable  and  stationary  ether. 

13.  We  have  thus  briefly  traced  the  development  of  the  ether 
theory  of  light  and  electromagnetism.  We  have  seen  that  the  space 
continuum  assumed  by  this  theory  is  not  empty  as  was  the  space  of 
Newton  and  Galileo  but  is  assumed  filled  with  a  stationary  medium, 
the  ether,  and  in  conclusion  should  further  point  out  that  the  time 
continuum  assumed  by  the  ether  theory  was  apparently  the  same  as 
that  of  Newton  and  Galileo,  and  in  particular  that  the  old  ideas  as  to 
the  absolute  independence  of  space  and  time  were  all  retained. 


Historical  Development.  17 

PART  III.     RISE  OF  THE  EINSTEIN  THEORY  OF  RELATIVITY. 

14.  The  Michelson-Morley  Experiment.  In  spite  of  all  the  brilli- 
ant achievements  of  the  theory  of  a  stationary  ether,  we  must  now 
call  attention  to  an  experiment,  performed  at  the  very  time  when 
the  success  of  the  ether  theory  seemed  most  complete,  whose  result 
was  in  direct  contradiction  to  its  predictions.  This  is  the  celebrated 
Michelson-Morley  experiment,  and  to  the  masterful  interpretation  of 
its  consequences  at  the  hands  of  Einstein  we  owe  the  whole  theory  of 
relativity,  a  theory  which  will  nevermore  permit  us  to  assume  that 
space  and  time  are  independent. 

If  the  theory  of  a  stationary  ether  were  true  we  should  find,  con- 
trary to  the  expectations  of  Newton,  that  systems  of  coordinates  in 
relative  motion  are  not  symmetrical,  a  system  of  axes  fixed  relatively 
to  the  ether  would  hold  a  unique  position  among  all  other  systems 
moving  relative  to  it  and  would  be  peculiarly  adapted  for  the  measure- 
ment of  displacements  and  velocities.  Bodies  at  rest  with  respect 
to  this  system  of  axes  fixed  in  the  ether  would  be  spoken  of  as  "  ab- 
solutely "  at  rest  and  bodies  in  motion  through  the  ether  would  be 
said  to  have  "  absolute  "  motion.  From  the  point  of  view  of  the 
ether  theory  one  of  the  most  important  physical  problems  would  be 
to  determine  the  velocity  of  various  bodies,  for  example  that  of  the 
earth,  through  the  ether. 

Now  the  Michelson-Morley  experiment  was  devised  for  the  very 
purpose  of  determining  the  relative  motion  of  the  earth  and  the  ether. 
The  experiment  consists  essentially  in  a  comparison  of  the  velocities 
of  light  parallel  and  perpendicular  to  the  earth's  motion  in  its  orbit. 
A  ray  of  light  from  the  source  S  falls  on  the  half  silvered  mirror  A, 
where  it  is  divided  into  two  rays,  one  of  which  travels  to  the  mirror  B 
and  the  other  to  the  mirror  C,  where  they  are  totally  reflected.  The 
rays  are  recombined  and  produce  a  set  of  interference  fringes  at  0. 
(See  figure  1.) 

We  may  now  think  of  the  apparatus  as  set  so  that  one  of  the 
divided  paths  is  parallel  to  the  earth's  motion  and  the  other  per- 
pendicular to  it.  On  the  basis  of  the  stationary  ether  theory,  the 
velocity  of  the  light  with  reference  to  the  apparatus  would  evidently 
be  different  over  the  two  paths,  and  hence  on  rotating  the  apparatus 

3 


18 


Chapter  One. 


through  an  angle  of  ninety  degrees  we  should  expect  a  shift  in  the 
position  of  the  fringes.  Knowing  the  magnitude  of  the  earth's 
velocity  in  its  orbit  and  the  dimensions  of  the  apparatus,  it  is  quite 
possible  to  calculate  the  magnitude  of  the  expected  shift,  a  quantity 


o 

FIG.  1. 

entirely  susceptible  of  experimental  determination.  Nevertheless  the 
most  careful  experiments  made  at  different  times  of  day  and  at 
different  seasons  of  the  year  entirely  failed  to  show  any  such  shift 
at  all. 

This  result  is  in  direct  contradiction  to  the  theory  of  a  stationary 
ether  and  could  be  reconciled  with  that  theory  only  by  very  arbitrary 
assumptions.  Instead  of  making  such  assumptions,  the  Einstein 
theory  of  relativity  finds  it  preferable  to  return  in  part  to  the  older 
ideas  of  Newton  and  Galileo. 

15.  The  Postulates  of  Einstein.  In  fact,  in  accordance  with  the 
results  of  this  work  of  Michelson-Morley  and  other  confirmatory 
experiments,  the  Einstein  theory  takes  as  its  first  postulate  the  idea 
familiar  to  Newton  of  the  relativity  of  all  motion.  It  states  that 
there  is  nothing  out  in  space  in  the  nature  of  an  ether  or  of  a  fixed 
set  of  coordinates  with  regard  to  which  motion  can  be  measured, 
that  there  is  no  such  thing  as  absolute  motion,  and  that  all  we  can 
speak  of  is  the  relative  motion  of  one  body  with  respect  to  another. 


Historical  Development.  19 

Although  we  thus  see  that  the  Einstein  theory  of  relativity  has 
returned  in  part  to  the  ideas  of  Newton  and  Galileo  as  to  the  nature 
of  space,  it  is  not  to  be  supposed  that  the  ether  theory  of  light  and 
electromagnetism  has  made  no  lasting  contribution  to  physical  science. 
Quite  on  the  contrary,  not  only  must  the  ideas  as  to  the  periodic  and 
polarizable  nature  of  the  light  disturbance,  which  were  first  appre- 
ciated and  understood  with  the  help  of  the  ether  theory,  always 
remain  incorporated  in  every  optical  theory,  but  in  particular  the 
Einstein  theory  of  relativity  takes  as  the  basis  for  its  second  postulate 
a  principle  that  has  long  been  familiar  to  the  ether  theory,  namely 
that  the  velocity  of  light  is  independent  of  the  velocity  of  the  source. 
We  shall  see  in  following  chapters  that  it  is  the  combination  of  this 
principle  with  the  first  postulate  of  relativity  that  leads  to  the  whole 
theory  of  relativity  and  to  our  new  ideas  as  to  the  nature  and  inter- 
relation of  space  and  time. 


CHAPTER  II. 

THE  TWO  POSTULATES  OF  THE  EINSTEIN  THEORY  OF 
RELATIVITY. 

16.  There  are  two  general  methods  of  evaluating  the  theoretical 
development  of  any  branch  of  science.     One  of  these  methods  is  to 
test  by  direct  experiment  the  fundamental  postulates  upon  which 
the  theory  rests.     If  these  postulates  are  found  to  agree  with  the  facts, 
we  may  feel  justified  in  assuming  that  the  whole  theoretical  structure 
is  a  valid  one,  providing  false  logic  or  unsuspected  and  incorrect 
assumptions  have  not  later  crept  in  to  vitiate  the  conclusions.     The 
other  method  of  testing  a  theory  is  to  develop  its  interlacing  chain  of 
propositions  and  theorems  and  examine  the  results  both  for  their 
internal  coherence  and  for  their  objective  validity.     If  we  find  that 
the  conclusions  drawn  from  the  theory  are  neither  self-contradictory 
nor  contradictory  of  each  other,  and  furthermore  that  they  agree 
with  the  facts  of  the  external  world,  we  may  again  feel  that  our  theory 
has  achieved  a  measure  of  success.     In  the  present  chapter  we  shall 
present  the  two  main  postulates  of  the  theory  of  relativity,  and  indicate 
the  direct  experimental  evidence  in  favor  of  their  truth.     In  following 
chapters  we  shall  develop  the  consequences  of  these  postulates,  show 
that  the  system  of  consequences  stands  the  test  of  internal  coherence, 
and  wherever  possible  compare  the  predictions  of  the  theory  with 
experimental  facts. 

The  First  Postulate  of  Relativity. 

17.  The  first  postulate  of  relativity  as  originally  stated  by  Newton 
was  that  it  is  impossible  to  measure  or  detect  absolute  translatory 
motion  through  space.     No  objections  have  ever  been  made  to  this 
statement  of  the  postulate  in  its  original  form.     In  the  development 
of  the  theory  of  relativity,  the  postulate  has  been  modified  to  include 
the  impossibility  of  detecting  translatory  motion  through  any  medium 
or  ether  which  might  be  assumed  to  pervade  space. 

In  support  of  the  principle  is  the  general  fact  that  no  effects  due 
to  the  motion  of  the  earth  or  other  body  through  the  supposed  ether 

20 


The  Two  Postulates.  21 

have  ever  been  observed.  Of  the  many  unsuccessful  attempts  to 
detect  the  earth's  motion  through  the  ether  we  may  call  attention  to 
the  experiments  on  the  refraction  of  light  made  by  Arago,  Respighi, 
Hoek,  Ketteler  and  Mascart,  the  interference  experiments  of  Ketteler 
and  Mascart,  the  work  of  Klinkerfuess  and  Haga  on  the  position  of 
the  absorption  bands  of  sodium,  the  experiment  of  Nordmeyer  on  the 
intensity  of  radiation,  the  experiments  of  Fizeau,  Brace  and  Strasser 
on  the  rotation  of  the  plane  of  polarized  light  by  transmission  through 
glass  plates,  the  experiments  of  Mascart  and  of  Rayleigh  on  the 
rotation  of  the  plane  of  polarized  light  in  naturally  active  substances, 
the  electromagnetic  experiments  of  Rontgen,  Des  Coudres,  J.  Koenigs- 
berger,  Trouton,  Trouton  and  Noble,  and  Trouton  and  Rankine,  and 
finally  the  Michelson  and  Morley  experiment,  with  the  further  work 
of  Morley  and  Miller.  For  details  as  to  the  nature  of  these  experi- 
ments the  reader  may  refer  to  the  original  articles  or  to  an  excellent 
discussion  by  Laub  of  the  experimental  basis  of  the  theory  of  rela- 
tivity. * 

In  none  of  the  above  investigations  was  it  possible  to  detect  any 
effect  attributable  to  the  earth's  motion  through  the  ether.  Never- 
theless a  number  of  these  experiments  are  in  accord  with  the  final 
form  given  to  the  ether  theory  by  Lorentz,  especially  since  his  work 
satisfactorily  accounts  for  the  Fresnel  coefficient  for  the  changed 
velocity  of  light  in  moving  media.  Others  of  the  experiments  men- 
tioned, however,  could  be  made  to  accord  withvthe  Lorentz  theory 
only  by  very  arbitrary  assumptions,  in  particular  those  of  Michelson 
and  Morley,  Mascart  and  Rayleigh,  and  Trouton  and  Noble.  For 
the  purposes  of  our  discussion  we  shall  accept  the  principle  of  the 
relativity  of  motion  as  an  experimental  fact. 

The  Second  Postulate  of  the  Einstein  Theory  of  Relativity. 

18.  The  second  postulate  of  relativity  states  that  the  velocity  of 
light  in  free  space  appears  the  same  to  all  observers  regardless  of  the 
relative  motion  of  the  source  of  light  and  the  observer.  This  postulate 
may  be  obtained  by  combining  the  first  postulate  of  relativity  with  a 
principle  which  has  long  been  familiar  to  the  ether  theory  of  light. 
This  principle  states  that  the  velocity  of  light  is  unaffected  by  a 
motion  of  the  emitting  source,  in  other  words,  that  the  velocity  with 

*  Jahrbuch  der  Radioaktiritat,  vol.  7,  p.  405  (1910). 


22  Chapter  Two. 

which  light  travels  past  any  observer  is  not  increased  by  a  motion 
of  the  source  of  light  towards  the  observer.  The  first  postulate  of 
relativity  adds  the  idea  that  a  motion  of  the  source  of  light  towards 
the  observer  is  identical  with  a  motion  of  the  observer  towards  the 
source.  The  second  postulate  of  relativity  is  seen  to  be  merely  a 
combination  of  these  two  principles,  since  it  states  that  the  velocity 
of  light  in  free  space  appears  the  same  to  all  observers  regardless  both 
of  the  motion  of  the  source  of  light  and  of  the  observer. 

19.  It  should  be  pointed  out  that  the  two  principles  whose  com- 
bination thus  leads  to  the  second  postulate  of  Einstein  have  come 
from  very  different  sources.  The  first  postulate  of  relativity  prac- 
tically denies  the  existence  of  any  stationary  ether  through  which 
the  earth,  for  instance,  might  be  moving.  On  the  other  hand,  the 
principle  that  the  velocity  of  light  is  unaffected  by  a  motion  of  the 
source  was  originally  derived  from  the  idea  that  light  is  transmitted 
by  a  stationary  medium  which  does  not  partake  in  the  motion  of  the 
source.  This  combination  of  two  principles,  which  from  a  historical 
point  of  view  seem  somewhat  contradictory  in  nature,  has  given  to 
the  second  postulate  of  relativity  a  very  extraordinary  content. 
Indeed  it  should  be  particularly  emphasized  that  the  remarkable 
conclusions  as  to  the  nature  of  space  and  time  forced  upon  science 
by  the  theory  of  relativity  are  the  special  product  of  the  second 
postulate  of  relativity. 

A  simple  example  of  the  conclusions  which  can  be  drawn  from 
this  postulate  will  make  its  extraordinary  nature  evident. 


a' 


b  B  b' 

FIG.  2. 


S  is  a  source  of  light  and  A  and  B  two  moving  systems.  A  is 
moving  towards  the  source  S,  and  B  away  from  it.  Observers  on  the 
systems  mark  off  equal  distances  aaf  and  W  along  the  path  of  the  light 
and  determine  the  time  taken  for  light  to  pass  from  a  to  a'  and  b  to  &' 
respectively.  Contrary  to  what  seem  the  simple  conclusions  of 
common  sense,  the  second  postulate  requires  that  the  time  taken 


The  Two  Postulates.  23 

for  the  light  to  pass  from  a  to  a!  shall  measure  the  same  as  the  time 
for  the  light  to  go  from  b  to  b'.  Hence  if  the  second  postulate  of 
relativity  is  correct  it  is  not  surprising  that  science  is  forced  in  general 
to  new  ideas  as  to  the  nature  of  space  and  time,  ideas  which  are  in 
direct  opposition  to  the  requirements  of  so-called  common  sense. 

Suggested  Alternative  to  the  Postulate  of  the  Independence  of  the 

Velocity  of  Light  and  the  Velocity  of  the  Source. 

20.  Because  of  the  extraordinary  conclusions  derived  by  com- 
bining the  principle  of  the  relativity  of  motion  with  the  postulate 
that  the  velocity  of  light  is  independent  of  the  velocity  of  its  source, 
a  number  of  attempts  have  been  made  to  develop  so-called  emission 
theories  of  relativity  based  on  the  principle  of  the  relativity  of  motion 
and  the  further  postulate  that  the  velocity  of  light  and  the  velocity 
of  its  source  are  additive. 

Before  examining  the  available  evidence  for  deciding  between  the 
rival  principles  as  to  the  velocity  of  light,  we  may  point  out  that 
this  proposed  postulate,  of  the  additivity  of  the  velocity  of  source 
and  light,  would  as  a  matter  of  fact  lead  to  a  very  simple  kind  of 
relativity  theory  without  requiring  any  changes  in  our  notions  of 
space  and  time.  For  if  light  or  other  electromagnetic  disturbance 
which  is  being  emitted  from  a  source  did  partake  in  the  motion  of 
that  source  in  such  a  way  that  the  velocity  of  the  source  is  added  to 
the  velocity  of  emission,  it  is  evident  that  a  system  consisting  of  the 
source  and  its  surrounding  disturbances  would  act  as  a  whole  and 
suffer  no  permanent  change  in  configuration  if  the  velocity  of  the 
source  were  changed.  This  result  would  of  course  be  in  direct  agree- 
ment with  the  idea  of  the  relativity  of  motion  which  merely  requires 
that  the  physical  properties  of  a  system  shall  be  independent  of  its 
velocity  through  space. 

As  a  particular  example  of  the  simplicity  of  emission  theories  we 
may  show,  for  instance,  how  easily  they  would  account  for  the  nega- 
tive result  of  the  Michelson-Morley  experiment.  If  0,  figure  3,  is  a 
source  of  light  and  A  and  B  are  mirrors  placed  a  meter  away  from  0,  the 
Michelson-Morley  experiment  shows  that  the  time  taken  for  light  to 
travel  to  A  and  back  is  the  same  as  for  the  light  to  travel  to  B  and 
back,  in  spite  of  the  fact  that  the  whole  apparatus  is  moving  through 
space  in  the  direction  0  —  B,  due  to  the  earth's  motion  around  the  sun. 


24  Chapter  Two. 

The  basic  assumption  of  emission  theories,  however,  would  require 
exactly  this  result,  since  it  says  that  light  travels  out  from  0  with  a 

constant  velocity  in  all  directions  with 
respect  to  0.  and  not  with  respect  to 
some  ether  through  which  0  is  supposed 
to  be  moving. 

Direction  of  Earth's  Motion  The   problem   now  before   us   is  to 

decide  between  the  two  rival  principles 
as  to  the  velocity  of  light,  and  we  shall 

' J  Ig          find  that  the  bulk  of  the  evidence  is  all 

FlG  3  in  favor  of  the  principle  which  has  led 

to  the  Einstein  theory  of  relativity  with 

its  complete  revolution  in  our  ideas  as  to  space  and  time,  and  against 
the  principle  which  has  led  to  the  superficially  simple  emission  theo- 
ries of  relativity. 

21.  Evidence  Against  Emission  Theories  of  Light.  All  emission 
theories  agree  in  assuming  that  light  from  a  moving  source  has  a 
velocity  equal  to  the  vector  sum  of  the  velocity  of  light  from  a  sta- 
tionary source  and  the  velocity  of  the  source  itself  at  the  instant  of 
emission.  And  without  first  considering  the  special  assumptions 
which  distinguish  one  emission  theory  from  another  we  may  first 
present  certain  astronomical  evidence  which  apparently  stands  in 
contradiction  to  this  basic  assumption  of  all  forms  of  emission 
theory.  This  evidence  was  pointed  out  by  Comstock*  and  later  by 
de  Sitter,  f 

Consider  the  rotation  of  a  binary  star  as  it  would  appear  to  an 
observer  situated  at  a  considerable  distance  from  the  star  and  in  its 
plane  of  rotation.  (See  figure  4.)  If  an  emission  theory  of  light 
be  true,  the  velocity  of  light  from  the  star  in  position  A  will  be  c  +  u, 
where  u  is  the  velocity  of  the  star  in  its  orbit,  while  in  the  position  B 
the  velocity  will  be  c  —  u.  Hence  the  star  will  be  observed  to  arrive 

I 

in  position  A,  seconds  after  the  event  has  actually  occurred,  and 

c  ~\~  u 

I 

in  position  B,  -      -  seconds  after  the  event  has  occurred.     This  will 

'  c  —  u 

*  Phys.  Rev.,  vol.  30,  p.  291  (1910). 

jPhys.  Zeitschr.,vo\.  14,  pp.  429,  1267  (1913). 


The  Two  Postulates.  25 

make  the  period  of  half  rotation  from  A  to  B  appear  to  be 


c  +  u      c  -  u  CL  ' 

where  A£  is  the  actual  time  of  a  half  rotation  in  the  orbit,  which  for 

AQ  U          > 

/"'         "N     ' 
/  \ 

j        o j z >o 

\  /  Observer 

\ 


(          O— :T. -0 

\  /  Observer 


\ 

\ 


FIG.  4. 

simplicity  may  be  taken  as  circular.     On  the  other  hand,  the  period 
of  the  next  half  rotation  from  B  back  to  A  would  appear  to  be 

2ul 
A*--T. 

2ul 

Xow  in  the  case  of  most  spectroscopic  binaries  the  quantity  — — 

c 

is  not  only  of  the  same  order  of  magnitude  as  AZ  but  oftentimes  prob- 
ably even  larger.  Hence,  if  an  emission  theory  of  light  were  true, 
we  could  hardly  expect  without  correcting  for  the  variable  velocity 
of  light  to  find  that  these  orbits  obey  Kepler's  laws,  as  is  actually 
the  case.  This  is  certainly  very  strong  evidence  against  any  form 
of  emission  theory.  It  may  not  be  out  of  place,  however,  to  state 
briefly  the  different  forms  of  emission  theory  which  have  been  tried. 
22.  Different  Forms  of  Emission  Theory.  As  we  have  seen,  emis- 
sion theories  all  agree  in  assuming  that  light  from  a  moving  source 


26  Chapter  Two. 

has  a  velocity  equal  to  the  vector  sum  of  the  velocity  of  light  from  a 
stationary  source  and  the  velocity  of  the  source  itself  at  the  instant 
of  emission.  Emission  theories  differ,  however,  in  their  assumptions 
as  to  the  velocity  of  light  after  its  reflection  from  a  mirror.  The  three 
assumptions  which  up  to  this  time  have  been  particularly  considered 
are  (1)  that  the  excited  portion  of  the  reflecting  mirror  acts  as  a  new 
source  of  light  and  that  the  reflected  light  has  the  same  velocity  c 
with  respect  to  the  mirror  as  has  original  light  with  respect  to  its  source ; 
(2)  that  light  reflected  from  a  mirror  acquires  a  component  of  velocity 
equal  to  the  velocity  of  the  mirror  image  of  the  original  source,  and 
hence  has  the  velocity  c  with  respect  to  this  mirror  image;  and  (3)  that 
light  retains  throughout  its  whole  path  the  component  of  velocity 
which  it  obtained  from  its  original  moving  source,  and  hence  after 
reflection  spreads  out  with  velocity  c  in  a  spherical  form  around  a 
center  which  moves  with  the  same  speed  as  the  original  source. 

Of  these  possible  assumptions  as  to  the  velocity  of  reflected  light, 
the  first  seems  to  be  the  most  natural  and  was  early  considered  by  the 
author  but  shown  to  be  incompatible,  not  only  with  an  experiment 
which  he  performed  on  the  velocity  of  light  from  the  two  limbs  of 
the  sun,*  but  also  with  measurements  of  the  Stark  effect  in  canal 
rays.f  The  second  assumption  as  to  the  velocity  of  light  was  made 
by  Stewart,  t  but  has  also  been  shown  f  to  be  incompatible  with 
measurements  of  the  Stark  effect  in  canal  rays.  Making  use  of  the 
third  assumption  as  to  the  velocity  of  reflected  light,  a  somewhat 
complete  emission  theory  has  been  developed  by  Ritz,§  and  un- 
fortunately optical  experiments  for  deciding  between  the  Einstein 
and  Ritz  relativity  theories  have  never  been  performed,  although 
such  experiments  are  entirely  possible  of  performance.!  Against  the 
Ritz  theory,  however,  we  have  of  course  the  general  astronomical 
evidence  of  Comstock  and  de  Sitter  which  we  have  already  described 
above. 

For  the  present,  the  observations  described  above,  comprise  the 
whole  of  the  direct  experimental  evidence  against  emission  theories 

*  Phys.  Rev.,  vol.  31,  p.  26  (1910). 
t  Phys.  Rev.,  vol.  35,  p.  136  (1912). 
t  Phys.  Rev.,  vol.  32,  p.  418  (1911). 

§  Arm.  de  chim.  et  phys.,  vol.  13,  p.  145  (1908);  Arch,  de  Geneve,  vol.  26,  p.  232 
(1908);  Scientia,  vol.  5  (1909). 


The  Two  Postulates.  27 

of  light  and  in  favor  of  the  principle  which  has  led  to  the  second 
postulate  of  the  Einstein  theory.  One  of  the  consequences  of  the 
Einstein  theory,  however,  has  been  the  deduction  of  an  expression 
for  the  mass  of  a  moving  body  which  has  been  closely  verified  by  the 
Kaufmann-Bucherer  experiment.  Now  it  is  very  interesting  to  note, 
that  starting  with  what  has  thus  become  an  experimental  expression 
for  the  mass  of  a  moving  body,  it  is  possible  to  work  backwards  to  a 
derivation  of  the  second  postulate  of  relativity.  For  the  details  of 
the  proof  we  must  refer  the  reader  to  the  original  article.* 

Further  Postulates  of  the  Theory  of  Relativity. 

23.  In  the  development  of  the  theory  of  relativity  to  which  we 
shall  now  proceed  we  shall  of  course  make  use  of  many  postulates. 
The  two  which  we  have  just  considered,  however,  are  the  only  ones, 
so  far  as  we  are  aware,  which  are  essentially  different  from  those 
common  to  the  usual  theoretical  developments  of  physical  science. 
In  particular  in  our  further  work  we  shall  assume  without  examination 
all  such  general  principles  as  the  homogeneity  and  isotropism  of  the 
space  continuum,  and  the  unidirectional,  one-valued,  one-dimensional 
character  of  the  time  continuum.  In  our  treatment  of  the  dynamics 
of  a  particle  we  shall  also  assume  Newton's  laws  of  motion,  and  the 
principle  of  the  conservation  of  mass,  although  we  shall  find,  of  course, 
that  the  Einstein  ideas  as  to  the  connection  between  space  and  time 
will  lead  us  to  a  non-Newtonian  mechanics.  We  shall  also  make 
extensive  use  of  the  principle  of  least  action,  which  we  shall  find  a 
powerful  principle  in  all  the  fields  of  dynamics. 

*  Phys.  Rev.,  vol.  31,  p.  26  (1910). 


CHAPTER  III. 
SOME  ELEMENTARY  DEDUCTIONS. 

24.  In  order  gradually  to  familiarize  the  reader  with  the  conse- 
quences of  the  theory  of  relativity  we  shall  now  develop  by  very 
elementary  methods  a  few  of  the  more  important  relations.     In  this 
preliminary  consideration  we  shall  lay  no  stress  on  mathematical 
elegance   or   logical   exactness.     It   is   believed,   however,   that   the 
chapter  will  present  a  substantially  correct  account  of  some  of  the 
more  important  conclusions  of  the  theory  of  relativity,  in  a  form 
which   can  be  understood  even  by  readers  without  mathematical 
equipment. 

Measurements  of  Time  in  a  Moving  System. 

25.  We  may  first  derive  from  the  postulates  of  relativity  a  relation 
connecting  measurements  of  time  intervals  as  made  by  observers  in 
systems    moving   with   different   velocities.     Consider   a   system   S 
(Fig.  5)  provided  with  a  plane  mirror  a  a,  and  an  observer  A,  who 


FIG.  5. 

has  a  clock  so  that  he  can  determine  the  time  taken  for  a  beam  of 
light  to  travel  up  to  the  mirror  and  back  along  the  path  Am  A. 
Consider  also  another  similar  system  £',  provided  with  a  mirror  b  6, 
and  an  observer  B,  who  also  has  a  clock  for  measuring  the  time  it 
takes  for  light  to  go  up  to  his  mirror  and  back.  System  Sf  is  moving 
past  S  with  the  velocity  V,  the  direction  of  motion  being  parallel 
to  the  mirrors  a  a  and  b  b}  the  two  systems  being  arranged,  more- 

28 


Some  Elementary  Deductions. 


29 


over,  so  that  when  they  pass  one  another  the  two  mirrors  a  a  and 
b  b  will  coincide,  and  the  two  observers  A  and  B  will  also  come  into 
coincidence. 

A,  considering  his  system  at  rest  and  the  other  in  motion,  measures 
the  time  taken  for  a  beam  of  light  to  pass  to  his  mirror  and  return, 
over  the  path  A  m  A,  and  compares  the  time  interval  thus  obtained 
with  that  necessary  for  the  performance  of  a  similar  experiment 
by  B,  in  which  the  light  has  to  pass  over  a  longer  path  such  as  B  n  Bf, 
shown  in  figure  6,  where  B  B'  is  the  distance  through  which  the 


/ 

*;  —          —  •• 

\ 

1 

\ 

I 

\ 

1 

\ 

• 

\ 

L        1 
1 

\ 

1 

\ 

1 
1 

\ 

B  -I 

i—  7?' 

i 

j                     U 

FIG. 

6. 

A 


observer  B  has  moved  during  the  time  taken  for  the  passage  of  the 
light  up  to  the  mirror  and  back. 

Since,  in  accordance  with  the  second  postulate  of  relativity,  the 
velocity  of  light  is  independent  of  the  velocity  of  its'  source,  it  is 
evident  that  the  ratio  of  these  two  time  intervals  will  be  proportional 
to  the  ratio  of  the  two  paths  A  m  A  and  B  n  B',  and  this  can  easily 
be  calculated  in  terms  of  the  velocity  of  light  c  and  the  velocity  V 
of  the  system  S'. 

From  figure  6  we  have 

(A  m)2  =  (p  n)2  =  (B  n)2  -  (B  p)2. 
Dividing  by  (B  n)2, 

(Am)2  (Bp)2 

(Bn)2   ''  (Bn)2' 

But  the  distance  B  p  is  to  B  n  as  V  is  to  c,  giving  us 

Am  _ 

Bn   '  ''   \       "  c2  ' 


30  Chapter  Three. 

and  hence  A  will  find,  either  by  calculation  or  by  direct  measurement 
if  he  has  arranged  clocks  at  B  and  B',  that  it  takes  a  longer  time  for 
the  performance  of  B's  experiment  than  for  the  performance  of  his 


/ 

:  \l  I  — 


own  in  the  ratio  1 

It  is  evident  from  the  first  postulate  of  relativity,  however,  that 
B  himself  must  find  exactly  the  same  length  of  time  for  the  light  to 
pass  up  to  his  mirror  and  come  back  as  did  A  in  his  experiment, 
because  the  two  systems  are,  as  a  matter  of  fact,  entirely  symmetrical 
and  we  could  with  equal  right  consider  B's  system  to  be  the  one  at 
rest  and  A's  the  one  in  motion. 

We  thus  find  that  two  observers,  A  and  B,  who  are  in  relative  motion 
will  not  in  general  agree  in  their  measurements  of  the  time  interval  neces- 
sary for  a  given  event  to  take  place,  the  event  in  this  particular  case, 
for  example,  having  been  the  performance  of  B's  experiment;  indeed, 
making  use  of  the  ratio  obtained  in  a  preceding  paragraph,  we  may 
go  further  and  make  the  quantitative  statement  that  measurements  of 
time  intervals  made  with  a  moving  clock  must  be  multiplied  by  the  quantity 

in  order  to  agree  with  measurements  made  with  a  stationary 

.       -viucj  c&o${\  re<j  em   *r 

system  of  clocks. 

It  is  sometimes  more  convenient  to  state  this  principle  in  the 
form:  A  stationary  observer  using  a  set  of  stationary  clocks  will 

obtain  a  greater  measurement  in  the  ratio  1 

time  interval  than  an  observer  who  uses  a  clock  moving  with  the 
velocity  V. 

Measurements  of  Length  in  a  Moving  System. 

26.  We  may  now  extend  our  considerations,  to  obtain  a  relation 
between  measurements  of  length  made  in  stationary  and  moving 
systems. 

As  to  measurements  of  length  perpendicular  to  the  line  of  motion 
of  the  two  systems  S  and  S',  a  little  consideration  will  make  it  at  once 
evident  that  both  A  and  B  must  obtain  identical  results.  This  is 
true  because  the  possibility  is  always  present  of  making  a  direct  com- 


Some  Elementary  Deductions.  31 

parison  of  the  meter  sticks  which  A  and  B  use  for  such  measurements 
by  holding  them  perpendicular  to  the  line  of  motion.  When  the 
relative  motion  of  the  two  systems  brings  such  meter  sticks  into 
juxtaposition,  it  is  evident  from  the  first  postulate  of  relativity  that 
A's  meter  and  B's  meter  must  coincide  in  length.  Any  difference  in 
length  could  be  due  only  to  the  different  velocity  of  the  two  systems 
through  space,  and  such  an  occurrence  is  ruled  out  by  our  first  postulate. 
Hence  measurements  made  with  a  moving  meter  stick  held  perpendicular 
to  its  line  of  motion  will  agree  with  those  made  with  stationary  meter 
sticks. 

27.  With  regard  to  measurements  of  length  parallel  to  the  line  of 
motion  of  the  systems,  the  affair  is  much  more  complicated.  Any 
direct  comparison  of  the  lengths  of  meter  sticks  in  the  two  systems 
would  be  very  difficult  to  carry  out;  the  consideration,  however,  of  a 
simple  experiment  on  the  velocity  of  light  parallel  to  the  motion  of 
the  systems  will  lead  to  the  desired  relation. 

Let  us  again  consider  two  systems  S  and  S'  (fig.  7),  Sr  moving 
past  S  with  the  velocity  V. 


m 


FIG.  7. 

A  and  B  are  observers  on  these  systems  provided  with  clocks  and 
meter  sticks.  The  two  observers  lay  off,  each  on  his  own  system, 
paths  for  measuring  the  velocity  of  light.  A  lays  off  a  distance  of 
one  meter,  A  m,  so  that  he  can  measure  the  time  for  light  to  travel 
to  the  mirror  m  and  return,  and  B,  using  a  meter  stick  which  has 
the  same  length  as  A's  when  they  are  both  at  rest,  lays  off  the  dis- 
tance B  n. 

Each  observer  measures  the  length  of  time  it  takes  for  light  to 
travel  to  his  mirror  and  return,  and  will  evidently  have  to  find  the 
same  length  of  time,  since  the  postulates  of  relativity  require  that  the 
velocity  of  light  shall  be  the  same  for  all  observers. 


32  Chapter  Three. 

Now  the  observer  A,  taking  himself  as  at  rest,  finds  that  B's 
light  travels  over  a  path  B  n'  B'  (fig.  8),  where  nn'  is  the  distance 


B 


B' 


FIG.  8. 

through  which  the  mirror  n  moves  while  the  light  is  travelling  up  to 
it,  and  B  B'  is  the  distance  through  which  the  source  travels  before 
the  light  gets  back.     It  is  easy  to  calculate  the  length  of  this  path. 
We  have 

nnf  _    V 

B  n'    |  c 
and 

BB'         V_ 
Bn'  B'  ~~~~c' 
Also,  from  the  figure, 

B  n'  =  B  n  -f  n  n', 
Bri  B'  =  BnB  +  2nn'  -  BB'. 

Combining,  we  ol 

Bn'  B'  _     _1_ 

B  n  W  :       "y* ' 

"  c8 

Thus  A  finds  that  the  path  traversed  by  B's  light,  instead  of  being 
exactly  two  meters  as  was  his  own,  will  be  longer  in  the  ratio    of 

f          yz\ 

1 :  (  1 £-  1  .     For  this  reason  A  is  rather  surprised  that  B  does 

not  report  a  longer  time  interval  for  the  passage  of  the  light  than  he 
himself  found.  He  remembers,  however,  that  he  has  already  found 
that  measurements  of  time  made  with  -a  moving  clock  must  be  multi- 
plied by  the  quantity  .—  --  in  order  to  agree  with  his  own,  and 


sees  that  this  will  account  for  part  of  the  discrepancy  between  the 
expected  and  observed  results.  To  account  for  the  remaining  dis- 
crepancy the  further  conclusion  is  now  obtained  that  measurements  of 


Some  Elementary  Deductions.  33 

length  made  with  a  moving  meter  stick,  parallel  to  its  motion,  must  be 

I         V2   . 
multiplied  by  the  quantity  \jl  —  —  in  order  to  agree  with  those  made 

in  a  stationary  system. 

In  accordance  with  this  principle,  a  stationary  observer  will 
obtain  a  smaller  measurement  for  the  length  of  a  moving  body  than 
will  an  observer  moving  along  with  the  object.  This  has  been  called 
the  Lorentz  shortening,  the  shortening  occurring  in  the  ratio 

*          v 

in  the  line  of  motion. 

The  Setting  of  Clocks  in  a  Moving  System. 

28.  It  will  be  noticed  that  in  our  considerations  up  to  this  point 
we  have  considered  cases  Where  only  a  single  moving  clock  was  needed 
in  performing  the  desired  experiment,  and  this  was  done  purposely, 
since  we  shall  find,  not  only  that  a  given  time  interval  measures 
shorter  on  a  moving  clock  than  on  a  system  of  stationary  clocks, 
but  that  a  system  of  moving  clocks  which  have  been  set  in  synchronism 
by  an  observer  moving  along  with  them  will  not  be  set  in  synchronism 
for  a  stationary  observer. 

Consider  again  two  systems  S  and  Sf  in  relative  motion  with  the 
velocity  V.  An  observer  A  on  system  S  places  two  carefully  com- 
pared clocks,  unit  distance  apart,  in  the  line  of  motion,  and  has  the 
time  on  each  clock  read  when  a  given  point  on  the  other  system 
passes  it.  An  observer  B  on  system  Sf  performs  a  similar  experiment. 
The  time  interval  obtained  in  the  two  sets  of  readings  must  be  the 
same,  since  the  first  postulate  of  relativity  obviously  requires  that  the 
relative  velocity  of  the  two  systems  V  shall  have  the  same  value  for 
both  observers. 

The  observer  A,  however,  taking  himself  as  at  rest,  and  familiar 
with  the  change  in  the  measurements  of  length  and  time  in  the  moving 
system  which  have  already  been  deduced,  expects  that  the  velocity 
as  measured  by  B  will  be  greater  than  the  value  that  he  himself 

obtains  in  the  ratio  -     — —  ,  since  any  particular  one  of  B's  clocks 


34  Chapter  Three. 

gives  a  shorter  value  for  a  given  time  interval  than  his  own,  while 
B's  measurements  of  the  length  of  a  moving  object  are  greater  than 


his  own,  each  by  the  factor  \/l  —  —  .     In  order  to  explain  the  actual 

result  of  B's  experiment  he  now  has  to  conclude  that  the  clocks  which 
for  B  are  set  synchronously  are  not  set  in  synchronism  for  himself. 

From  what  has  preceded  it  is  easy  to  see  that  in  the  moving  system, 
from  the  point  of  view  of  the  stationary  observer,  clocks  must  be  set 
further  and  further  ahead  as  we  proceed  towards  the  rear  of  the 
system,  since  otherwise  B  would  not  obtain  a  great  enough  difference 
in  the  readings  of  the  clocks  as  they  come  opposite  the  given  point 
on  the  other  system.  Indeed,  if  two  clocks  are  situated  in  the  moving 
system,  S',  one  in  front  of  the  other  by  the  distance  I',  as  measured 
by  B}  then  for  A  it  will  appear  as  though  B  had  set  his  rear  clock  ahead 

I'V 

by  the  amount  —  . 

29.  We  have  now  obtained  all  the  information  which  we  shall 
need  in  this  chapter  as  to  measurements  of  time  and  length  in  systems 
moving  with  different  velocities.  We  may  point  out,  however,  before 
proceeding  to  the  application  of  these  considerations,  that  our  choice 
of  A's  system  as  the  one  which  we  should  call  stationary  was  of  course 
entirely  arbitrary  and  immaterial.  We  can  at  any  time  equally  well 
take  B's  system  as  the  one  to  which  we  shall  ultimately  refer  all  our 
measurements,  and  indeed  all  that  we  shall  mean  when  we  call  one  of 
our  systems  stationary  is  that  for  reasons  of  convenience  we  have 
picked  out  that  particular  system  as  the  one  with  reference  to  which 
we  particularly  wish  to  make  our  measurements.  We  may  also 
point  out  that  of  course  B  has  to  subject  A's  measurements  of  time 

and  length  to  just  the  same  multiplications  by  the  factor  —  1= 


I  V2 

V1-  7 


c' 
as  did  A  in  order  to  make  them  agree  with  his  own. 

These  conclusions  as  to  measurements  of  space  and  time  are  of  course 
very  startling  when  first  encountered.  The  mere  fact,  however,  that 
they  appear  strange  to  so-called  "  common  sense  "  need  cause  us 
no  difficulty,  since  the  older  ideas  of  space  and  time  were  obtained 
from  an  ancestral  experience  which  never  included  experiments  with 


Some  Elementary  Deductions.  35 

72 
high  relative  velocities,  and  it  is  only  when  the  ratio   —  becomes 

appreciable  that  we  obtain  unexpected  results.  To  those  scientists 
who  do  not  wish  to  give  up  their  "  common  sense  "  ideas  of  space 
and  time  we  can  merely  say  that  if  they  accept  the  two  postulates 
of  relativity  then  they  will  also  have  to  accept  the  consequences 
which  can  be  deduced  therefrom.  The  remarkable  nature  of  these 
consequences  merely  indicates  the  very  imperfect  nature  of  our  older 
conceptions  of  space  and  time. 

The  Composition  of  Velocities. 

30.  Our  conclusions  as  to  the  setting  of  clocks  make  it  possible 
to  obtain  an  important  expression  for  the  composition  of  velocities. 
Suppose  we  have  a  system  S,  which  we  shall  take  as  stationary,  and 
on  the  system  an  observer  A.  Moving  past  S  with  the  velocity  V 
is  another  system  S'  with  an  observer  B,  and  finally  moving  past  S' 
in  the  same  direction  is  a  body  whose  velocity  is  u'  as  measured  by 
observer  B.  What  will  be  the  velocity  u  of  this  body  as  measured 
by  A? 

Our  older  ideas  led  us  to  believe  in  the  simple  additivity  of  veloci- 
ties and  we  should  have  calculated  u  in  accordance  with  the  simple 
expression 

u  =  V  +  u'. 

We  must  now  allow,  however,  for  the  fact  that  u'  is  measured  with 
clocks  which  to  A  appear  to  be  set  in  a  peculiar  fashion  and  running 
at  a  different  rate  from  his  own,  and  with  meter  sticks  which  give 
longer  measurements  than  those  used  in  the  stationary  system. 

The  determination  of  u'  by  observer  B  would  be  obtained  by 
measuring  the  time  interval  necessary  for  the  body  in  question  to 
move  a  given  distance  I'  along  the  system  S'.  If  t'  is  the  difference 
in  the  respective  clock  readings  when  the  body  reaches  the  ends  of 
the  line  lf,  we  have 

*-?-. 

I'V 
We  have  already  seen,  however,  that  the  two  clocks  are  for  A  set  - 

c2 

units  apart  and  hence  for  clocks  set  together  the  time  interval  would 


36  Chapter  Three, 

rv 

have  measured  t'  -\ ;r .     Furthermore    these    moving    clocks    give 

c 


time  measurements  which  are  shorter  in  the  ratio  \/l  —  —  :  1   than 


/  V2 

10    >Jl   - 


those  obtained  by  A,  so  that  for  A  the  time  interval  for  the  body  to 
move  from  one  end  of  I'  to  the  other  would  measure 


1  - 


furthermore,  owing  to  the  difference  in  measurements  of  length,  this 

/         V2 
line  I'  has  for  A  the  length  I'  \l  1  —  —  . 

*  C 

body  is  moving  past  S*  with  the  velocity, 


/         F2 
line  V  has  for  A  the  length  I'  \j  1  —  —  .     Hence    A    finds    that    the 


c- 
This  makes  the  total  velocity  of  the  body  past  S  equal  to  the  sum 


u=-V+-  u'V     ' 

1  +- 


or 

V 


u  = 


u'V 

T       9" 


This  new  expression  for  the  composition  of  velocities  is  extremely 
important.  When  the  velocities  u'  and  V  are  small  compared  with 
the  velocity  of  light  c,  we  observe  that  the  formula  reduces  to  the  simple 
additivity  principle  which  we  know  by  common  experience  to  be  true 


Some  Elementary  Deductions.  37 

for  all  ordinary  velocities.  Until  very  recently  the  human  race  has 
had  practically  no  experience  with  high  velocities  and  we  now  see 
that  for  velocities  in  the  neighborhood  of  that  of  light,  the  simple 
additivity  principle  is  nowhere  near  true. 

In  particular  it  should  be  noticed  that  by  the  composition  of 
velocities  which  are  themselves  less  than  that  of  light  we  can  never 
obtain  any  velocity  greater  than  that  of  light.  As  an  extreme  case, 
suppose  for  example  that  the  system  S'  were  moving  past  S  itself 
with  the  velocity  of  light  (i.  e.,  V  =  c)  and  that  in  the  system  S'  a 
particle  should  itself  be  given  the  velocity  of  light  in  the  same  direc- 
tion (i.  e.,  u'  =  c);  we  find  on  substitution  that  the  particle  still  has 
only  the  velocity  of  light  with  respect  to  S.  We  have 

c  +  c       2c 


By  the  consideration  of  such  conclusions  as  these  the  reader  will 
appreciate  more  and  more  the  necessity  of  abandoning  his  older 
naive  ideas  of  space  and  time  which  are  the  inheritance  of  a  long 
human  experience  with  physical  systems  in  which  only  slow  velocities 
were  encountered. 

The  Mass  of  a  Moving  Body. 

31.  We  may  now  obtain  an  important  relation  for  the  mass  of  a 
moving  body.  Consider  again  two  similar  systems,  S  at  rest  and  S' 
moving  past  with  the  velocity  V.  The  observer  A  on  system  S  has  a 
sphere  made  from  some  rigid  elastic  material,  having  a  mass  of  m 
grams,  and  the  observer  B  on  system  S'  is  also  provided  with  a  similar 
sphere.  The  two  spheres  are  made  so  that  they  are  exactly  alike 
when  both  are  at  rest;  thus  B's  sphere,  since  it  is  at  rest  with  respect 
to  him,  looks  to  him  just  the  same  as  the  other  sphere  does  to  A- 
As  the  two  systems  pass  each  other  (fig.  9)  each  of  these  clever  experi- 
menters rolls  his  sphere  towards  the  other  system  with  a  velocity  of 
u  cm.  per  second,  so  that  they  will  just  collide  and  rebound  in  a  line 
perpendicular  to  the  direction  of  motion.  Now,  from  the  first  postu- 
late of  relativity,  system  S'  appears  to  B  just  the  same  as  system  S 
appears  to  A,  and  B's  ball  appears  to  him  to  go  through  the  same 
evolutions  that  A  finds  for  his  ball.  A  finds  that  his  ball  on  collision 


38 


Chapter  Three. 


undergoes  the  algebraic  change  of  velocity  2u,  B  finds  the  same  change 
in  velocity  2u  for  his  ball.  B  reports  this  fact  to  A,  and  A  knowing 
that  B's  measurements  of  length  agree  with  his  own  in  this  transverse 


FIG.  9. 


direction,  but  that  his  clock  gives  time  intervals  that  are  shorter  than 


his  own  in  the  ratio  \]1     -  —  :  1,  calculates  that  the  change  in  veloc- 


/         72 
ity  of  B's  ball  must  be  2u  -\J1  -  -  —  . 


c2 

From  the  principle  of  the  conservation  of  momentum,  however, 
A  knows  that  the  change  in  momentum  of  B's  ball  must  be  the  same 
as  that  of  his  own  and  hence  can  write  the  equation 


mau 


where  ma  is  the  mass  of  A's  ball  and  nib  is  the  mass  of  B's  ball.     Solv 
ing  we  have 

ma 


nib  = 


72 


In  other  words,  B's  ball,  which  had  the  same  mass  ma  as  A's  when 


Some  Elementary  Deductions.  39 

both  were  at  rest,  is  found  to  have  the  larger  mass       __  .      -  when 


placed  in  a  system  that  is  moving  with  the  velocity  V.* 

The  theory  of  relativity  thus  leads  to  the  general  expression 


Win 

m  = 


r™5j 

V1  -  ? 


for  the  mass  of  a  body  moving  with  the  velocity  u  and  having  the 
mass  ra0  when  at  rest. 

Since  we  have  very  few  velocities  comparable  with  that  of  light 

/         tf 
it  is  obvious  that  the  quantity  A/1  -  •  —  seldom  differs  much  from 

unity,  which  makes  the  experimental  verification  of  this  expression 
difficult.  In  the  case  of  electrons,  however,  which  are  shot  off  from 
radioactive  substances,  or  indeed  in  the  case  of  cathode  rays  produced 
with  high  potentials,  we  do  have  particles  moving  with  velocities 
comparable  to  that  of  light,  and  the  experimental  work  of  Kaufmann, 
Bucherer,  Hupka  and  others  in  this  field  provides  one  of  the  most 
striking  triumphs  of  the  theory  of  relativity. 

The  Relation  Between  Mass  and  Energy. 

32.  The  theory  of  relativity  has  led  to  very  important  conclusions 
as  to  the  nature  of  mass  and  energy.  In  fact,  we  shall  see  that  matter 
and  energy  are  apparently  different  names  for  the  same  fundamental 
entity. 

When  we  set  a  body  in  motion  it  is  evident  from  the  previous 
section  that  we  increase  -both-  its  mass, -as^wett  as  its-energy.  Now 
we  can  show  that  there  is  a  definite  ratio  between  the  amount  of 
energy  that  we  give  to  the  body  and  the  amount  of  mass  that  we 
give  to  it. 

If  the  force  /  acts  on  a  particle  which  is  free  to  move,  its  increase  in 
kinetic  energy  is  evidently 

AE  =  ffdl. 

But  the  force  acting,  is  by  definition,  equal  to  the  rate  of  increase  in 

*  In  carrying  out  this  experiment  the  transverse  velocities  of  the  balls  should 
be  made  negligibly  small  in  comparison  with  the  relative  velocity  of  the  systems  V. 


40  Chapter  Three. 

the  momentum  of  the  particle 

/  =  I  (mu). 
Substituting  we  have 


We  have,  however,  from  the  previous  section, 

mo 


m  = 


which,  solved  for  u,  gives  us 

r  m* 

u  =  c\/l  -  —  . 
\          m2 

Substituting  this  value  of  u  in  our  equation  for  AE  we  obtain,  after 
simplification, 

&E  =  fczdm  =  c2Aw. 

This  says  that  the  increase  of  the  kinetic  energy  of  the  particle, 
in  ergs,  is  equal  to  the  increase  in  mass,  in  grams,  multiplied  by  the 
square  of  the  velocity  of  light.  If  now  we  bring  the  particle  to  rest 
it  will  give  up  both  its  kinetic  energy  and  its  excess  mass.  Accepting 
the  principles  of  the  conservation  of  mass  and  energy,  we  know,  how- 
ever, that  neither  this  energy  nor  the  mass  has  been  destroyed;  they 
have  merely  been  passed  on  to  other  bodies.  There  is,  moreover, 
every  reason  to  believe  that  this  mass  and  energy,  which  were  asso- 
ciated together  when  the  body  was  in  motion  and  left  the  body  when 
it  was  brought  to  rest,  still  remain  always  associated  together.  For 
example,  if  the  body  should  be  brought  to  rest  by  setting  another 
body  into  motion,  it  is  of  course  a  necessary  consequence  of  our  con- 
siderations that  the  kinetic  energy  and  the  excess  mass  both  pass 
on  together  to  the  new  body  which  is  set  in  motion.  A  similar  con- 
clusion would  be  true  if  the  body  is  brought  to  rest  by  frictional  forces, 
since  the  heat  produced  by  the  friction  means  an  increase  in  the  kinetic 
energies  of  ultimate  particles. 


Some  Elementary  Deductions.  41 

In  general  we  shall  find  it  pragmatic  to  consider  that  matter  and 
energy  are  merely  different  names  for  the  same  fundamental  entity. 
One  gram  of  matter  is  equal  to  1021  ergs  of  energy. 

c2  =  (2.9986  X  1010)2  =  approx.  1021. 

This  apparently  extraordinary  conclusion  is  in  reality  one  which 
produces  the  greatest  simplification  in  science.  Not  to  mention 
numerous  special  applications  where  this  principle  is  useful,  we  may 
call  attention  to  the  fact  that  the  great  laws  of  the  conservation  of 
mass  and  of  energy  have  now  become  identical.  We  may  also  point 
out  that  those  opposing  camps  of  philosophic  materialists  who  defend 
matter  on  the  one  hand  or  energy  on  the  other  as  the  fundamental 
entity  of  the  universe  may  now  forever  cease  their  unimportant  bicker- 
ings. 


CHAPTER   IV. 


THE  EINSTEIN  TRANSFORMATION  EQUATIONS  FOR  SPACE 

AND  TIME. 

The  Lorentz  Transformation. 

33.  We  may  now  proceed  to  a  systematic  study  of  the  consequences 
of  the  theory  of  relativity. 

The  fundamental  problem  that  first  arises  ••  in  considering 
spatial  and  temporal  measurements  is  that  of  transforming  the 
description  of  a  given  kinematical  occurrence  from  the  variables  of 
one  system  of  coordinates  to  those  of  another  system  which  is  in 
motion  relative  to  the  first. 

Consider  two  systems  of  right-angled  Cartesian  coordinates  S 
and  S'  (fig.  10)  in  relative  motion  in  the  X  direction  with  the  velocity  V. 


-X 


FIG.  10. 


-X' 


V 


The  position  of  any  given  point  in  space  can  be  determined  by  speci- 
fying its  coordinates  x,  y,  and  z  with  respect  to  system  S  or  its  coordi- 
nates x1 ',  if  and  z1  with  respect  to  system  S'.  Furthermore,  for  the 
purpose  of  determining  the  time  at  which  any  event  takes  place,  we 
may  think  of  each  system  of  coordinates  as  provided  with  a  whole 
series  of  clocks  placed  at  convenient  intervals  throughout  the  system, 
the  clocks  of  each  series  being  set  and  regulated*  by  observers  in  the 

*  We  may  think  of  the  clocks  as  being  set  in  any  of  the  ways  that  are  usual 
in  practice.  Perhaps  the  simplest  is  to  consider  the  clocks  as  mechanisms  which 
have  been  found  to  "keep  time"  when  they  are  all  together  where  they  can  be 
examined  by  one  individual  observer.  The  assumption  can  then  be  made,  in  ac- 

42 


Transformation  Equations  for  Space  and  Time.  43 

corresponding  system.  The  time  at  which  the  event  in  question 
takes  place  may  be  denoted  by  t  if  determined  by  the  clocks  belonging 
to  system  S  and  by  t'  if  determined  by  the  clocks  of  system  S'. 

For  convenience  the  two  systems  S  and  S'  are  chosen  so  that  the 
axes  OX  and  O'X'  lie  in  the  same  line,  and  for  further  simplification 
we  choose,  as  our  starting-point  for  time  measurements,  t  and  tf  both 
equal  to  zero  when  the  two  origins  come  into  coincidence. 

The  specific  problem  now  before  us  is  as  follows:  If  a  given  kine- 
matical  occurrence  has  been  observed  and  described  in  terms  of  the 
variables  x',  y',  %'  and  t',  what  substitutions  must  we  make  for  the 
values  of  these  variables  in  order  to  obtain  a  correct  description  of  the 
same  kinematical  event  in  terms  of  the  variables  x,  y,  z  and  t?  In 
other  words,  we  want  to  obtain  a  set  of  transformation  equations 
from  the  variables  of  system  S'  to  those  of  system  S.  The  equations 
which  we  shall  present  were  first  obtained  by  Lorentz,  and  the  process 
of  changing  from  one  set  of  variables  to  the  other  has  generally  been 
called  the  Lorentz  transformation.  The  significance  of  these  equa- 
tions from  the  point  of  view  of  the  theory  of  relativity  was  first  appre- 
ciated by  Einstein. 

Deduction  of  the  Fundamental  Transformation  Equations. 

34.  It  is  evident  that  these  transformation  equations  are  going 
to  depend  on  the  relative  velocity  V  of  the  two  systems,  so  that  we 
may  write  for  them  the  expressions 

x'  =  F,(Vt  x,  y,  z,  0, 
y'  =  F*(V,  x,  y,  z,  0, 
z'  =  F*(V,  x,  ?/,  z,  0, 
t'  =  F*(V,  x,  y,  z,  t), 

where  FI,  F2,  etc.,  are  the  unknown  functions  whose  form  we  wish 
to  determine. 

It  is  possible  at  the  outset,  however,  greatly  to  simplify  these 
relations.  If  we  accept  the  idea  of  the  homogeneity  of  space  it  is 
evident  that  any  other  line  parallel  to  OXX'  might  just  as  well  have 
been  chosen  as  our  line  of  X-axes,  and  hence  our  two  equations  for 
x'  and  t'  must  be  independent  of  y  and  z.  Moreover,  as  to  the  equa- 

cordance  with  our  ideas  of  the  homogeneity  of  space,  that  they  will  continue  to 
"keep  time"  after  they  have  been  distributed  throughout  the  system. 


44  Chapter  Four. 

tions  for  y'  and  z'  it  is  at  once  evident  that  the  only  possible  solutions 
are  y'  =  y  and  z'  =  z.  This  is  obvious  because  a  meter  stick  held 
in  the  system  Sf  perpendicular  to  the  line  of  relative  motion,  OX', 
of  the  system  can  be  directly  compared  with  meter  sticks  held  similarly 
in  system  S,  and  in  accordance  with  the  first  postulate  of  relativity 
they  must  agree  in  length  in  order  that  the  systems  may  be  entirely 
symmetrical.  We  may  now  rewrite  our  transformation  equations 
in  the  simplified  form 

x'  =  Fi(7,  t,  x), 

y'  =  y, 

z'  =  z, 

t'  -  Ft(V,  t,  x), 

and  have  only  two  functions,  Fi  and  Fz,  whose  form  has  to  be  de- 
termined. 

To  complete  the  solution  of  the  problem  we  may  make  use  of  three 
further  conditions  which  must  govern  the  transformation  equations. 

35.  Three  Conditions  to  be  Fulfilled.     In  the  first  place,  when  the 
velocity  V  between  the  systems  is  small,  it  is  evident  that  the  trans- 
formation equations  must  reduce  to  the  form  that  they  had  in  New- 
tonian mechanics,  since  we  know  both  from  measurements  and  from 
everyday  experience  that  the  Newtonian  concepts  of  space  and  time 
are  correct  as  long  as  we  deal  with  slow  velocities.     Hence  the  limiting 
form  of  the  equations  as  V  approaches  zero  will  be  (cf.  Chapter  I, 
equations  3-4-5-6) 

x'  =  x  -  Vt, 

yf  =  y, 

z'  =  z, 

t'  =  t. 

36.  A  second  condition  is  imposed  upon  the  form  of  the  functions 
Fi  and  F2  by  the  first  postulate  of  relativity,  which  requires  that  the 
two  systems  S  and  S'  shall  be  entirely  symmetrical.     Hence  the 
transformation  equations  for  changing  from  the  variables  of  system  S 
to  those  of  system  S'  must  be  of  exactly  the  same  form  as  those  used 
in  the  reverse  transformation,  containing,  however,    —  V  wherever 
+  V  occurs  in  the  latter  equations.     Expressing  this  requirement  in 


Transformation  Equations  for  Space  and  Time.  45 

mathematical  form,  we  may  write  as  true  equations 

X  =  F,(-  y,*',*'), 

*-F,(-  7,  «',*'), 

where  FI  and  F2  must  have  the  same  form  as  above. 

37.  A  final  condition  is  imposed  upon  the  form  of  FI  and  F2  by 
the  second  postulate  of  relativity,  which  states  that  the  velocity  of  a 
beam  of  light  appears  the  same  to  all  observers  regardless  of  the 
motion  of  the  source  of  light  or  of  the  observer.     Hence  our  trans- 
formation equations  must  be  of  such  a  form  that  a  given  beam  of 
light  has  the  same  velocity,  c,  when  measured  in  the  variables  of  either 
system.     Let  us  suppose,  for  example,  that  at  the  instant  t  —  t'  =  0, 
when  the  two  origins  come  into  coincidence,  a  light  impulse  is  started 
from  the  common  point  occupied  by  0  and  0'.     Then,  measured  in 
the  coordinates  of  either  system,  the  optical  disturbance  which  is 
generated  must  spread  out  from  the  origin  in  a  spherical  form  with 
the  velocity  c.     Hence,  using  the  variables  of  system  S,  the  coordinates 
of  any  point  on  the  surface  of  the  disturbance  will  be  given  by  the 
expression 

&  +  y2  +  *  =  cH\  (7) 

while  using  the  variables  of  system  S'  we  should  have  the  similar 
expression 

z'2  +  2/'2  +  z'2  =  <?V*.  (8) 

Thus  we  have  a  particular  kinematical  occurrence,  the  spreading  out 
of  a  light  disturbance,  whose  description  is  known  in  the  variables 
of  either  system,  and  our  transformation  equations  must  be  of  such 
a  form  that  their  substitution  will  change  equation  (8)  to  equation  (7) . 
In  other  words,  the  expression  x2  +  y2  +  &  —  c2t2  is  to  be  an  invariant 
for  the  Lorentz  transformation. 

38.  The  Transformation  Equations.     The  three  sets  of  conditions 
which,  as  we  have  seen  in  the  last  three  paragraphs,  are  imposed  upon 
the  form  of  FI  and  F2  are  sufficient  to  determine  the  solution  of  the 
problem.     The  natural  method  of  solution  is  obviously  that  of  trial, 


46  Chapter  Four. 

and  we  may  suggest  the  solution  : 


(*  ~  Vt)  =  K(X  -  70,  (9) 


y'  =  y,  UO) 

*'  =  z,  (11) 

(12) 


where  we  have  placed  K  to  represent  the  important  and  continually 

1 

recurring  quantity 


It  will  be  found  as  a  matter  of  fact  by  examination  that  these 
solutions  do  fit  all  three  requirements  which  we  have  stated.  Thus, 
when  V  becomes  small  compared  with  the  velocity  of  light,  c,  the 
equations  do  reduce  to  those  of  Galileo  and  Newton.  Secondly,  if 
the  equations  are  solved  for  the  unprimed  quantities  in  terms  of  the 
primed,  the  resulting  expressions  have  an  unchanged  form  except  for 
the  introduction  of  -  -  V  in  place  of  +  V,  thus  fulfilling  the  require- 
ments of  symmetry  imposed  by  the  first  postulate  of  relativity.  And 
finally,  if  we  substitute  the  expressions  for  x',  y',  z'  and  t'  in  the  poly- 
nomial x'2  +  y'2  +  z'2  =  cH'2,  we  shall  obtain  the  expression  x2  +  y2 
-f  z2  —  c2t2  and  have  thus  secured  the  in  variance  of  x2  +  y1  +  z2  —  c2t2 
which  is  required  by  the  second  postulate  of  relativity. 

We  may  further  point  out  that  the  whole  series  of  possible  Lorentz 
transformations  form  a  group  such  that  the  result  of  two  successive 
transformations  could  itself  be  represented  by  a  single  transformation 
provided  we  picked  out  suitable  magnitudes  and  directions  for  the 
velocities  between  the  various  systems. 

It  is  also  to  be  noted  that  the  transformation  becomes  imaginary 
for  cases  where  V  >  c,  and  we  shall  find  that  this  agrees  with  ideas 
obtained  in  other  ways  as  to  the  speed  of  light  being  an  upper  limit 
for  the  magnitude  of  all  velocities. 


Transformation  Equations  for  Space  and  Time.  47 

Further  Transformation  Equations. 

39.  Before  making  any  applications  of  our  equations  we  shall  find 
it  desirable  to  obtain  by  simple  substitutions  and  differentiations  a 
series  of  further  transformation  equations  which  will  be  of  great  value 
in  our  future  work. 

By  the  simple  differentiation  of  equation  (12)  we  can  obtain 

dx 
where  we  have  put  x  for  -7-  . 

40.  Transformation  Equations  for  Velocity.     By  differentiation  of 
the  equations  for  x'  ',  y'  and  z',  nos.  (9),  (10)  and  (11),  and  substitution 

of  the  value  just  found  for  -7-  we  may  obtain  the  following  transfor- 


mation equations  for  velocity: 
*-V 


-I  1 

•    ~^~  C2 

where  the  placing  of  a  dot  has  the  familiar  significance  of  differentiation 

dx  dx' 

with  respect  to  time,  —  being  represented  by  x  and  —  by  xf. 

The  significance  of  .these  equations  for  the  transformation  of 
velocities  is  as  follows :  If  for  an  observer  in  system  S  a  point  appears 
to  be  moving  with  the  uniform  velocity  (x,  y,  z)  its  velocity  (x',  y',  z'), 
as  measured  by  an  observer  in  system  S',  is  given  by  these  expressions 

(14),  (15)  and  (16). 

41.  Transformation  Equations  for  the  Function — j=     •=.     These 


48  Chapter  Four. 

three  transformation  equations  for  the  velocity  components  of  a  point, 
permit  us  to  obtain  a  further  transformation  equation  for  an  important 
function  of  the  velocity  which  we  shall  find  continually  recurring  in 

our  later  work.      This  is  the  function  —  .  ,  where  we  have  indi- 


cated  the  total  velocity  of  the  point  by  u,  according  to  the  expression 
W2  =  ±2  +  2/2  +  22.  By  the  substitution  of  equations  (14),  (15)  and 
(16)  we  obtain  the  transformation  equation 


(17) 


42.  Transformation  Equations  for  Acceleration.  By  further  dif- 
ferentiating equations  (14),  (15)  and  (16)  and  simplifying,  we  easily 
obtain  three  new  equations  for  transforming  measurements  of  accel- 
eration from  system  Sf  to  S,  viz.  : 

(rV\~3 
1--~J     rt,  (18) 

(f 
l- 


or 


(11  y\-2  v  /        11  F 

i  -  ^r  )    ^y  +  "4  (  l  -  ^ 

--3^,        (20) 


CHAPTER  V. 
KINEMATICAL  APPLICATIONS. 

43.  The  various  transformation  equations  for  spatial  and  temporal 
measurements  which  were  derived  in  the  previous  chapter  may  now  be 
used  for  the  treatment  of  a  number  of  kinematical  problems.     In 
particular  it  will  be  shown  in  the  latter  part  of  the  chapter  that  a 
number  of  optical  problems  can  be  handled  with  extraordinary  facility 
by  the  methods  now  at  our  disposal. 

The  Kinematical  Shape  of  a  Rigid  Body. 

44.  We  may  first  point  out  that  the  conclusions  of  relativity  theory 
lead  us  to  quite  new  ideas  as  to  what  is  meant  by  the  shape  of  a  rigid 
body.     We  shall  find  that  the  shape  of  a  rigid  body  will  depend  entirely 
upon  the  relative  motion  of  the  body  and  the  observer  who  is  making 
measurements  on  it. 

Consider  a  rigid  body  which  is  at  rest  with  respect  to  system  S'. 
Let  Xi,  yi,  z\  and  xz',  2/2',  z%  be  the  coordinates  in  system  Sf  of  two 
points  in  the  body.  The  coordinates  of  the  same  points  as  measured 
in  system  S  can  be  found  from  transformation  equations  (9),  (10) 
and  (11),  and  by  subtraction  we  can  obtain  the  following  expressions 


-(x»'-Xir)l  (21) 

Oh  -  2/i)  =  (?»'  -  Vi1),  (22) 

(««-£»)  =  (*/  -  21'),  (23) 

3 

connecting  the  distances  between  the  pair  of  points  as  viewed  in  the 
two  systems.  In  making  this  subtraction  terms  containing  t  have 
been  cancelled  out  since  we  are  interested  in  the  simultaneous  positions 
of  the  points.  In  accordance  with  these  equations  we  may  distinguish 
then  between  the  geometrical  shape  of  a  body,  which  is  the  shape  that 
it  has  when  measured  on  a  system  of  axes  which  are  at  rest  relative 
to  it,  and  its  kinematical  shape,  which  is  given  by  the  coordinates  which 
5  49 


50  Chapter  Five. 

express  the  simultaneous  positions  of  its  various  points  when  it  is  in 
motion  with  respect  to  the  axes  of  reference.  We  see  that  the  kine- 
matical  shape  of  a  rigid  body  differs  from  its  geometrical  shape  by  a 
shortening  of  all  its  dimensions  in  the  line  of  motion  in  the  ratio 


1  --  g  :  1;  thus  a  sphere,  for  example,  becomes  a  Heaviside  ellipsoid. 

In  order  to  avoid  incorrectness  of  speech  we  must  be  very  care- 
ful not  to  give  the  idea  that  the  kinematical  shape  of  a  body  is  in 
any  sense  either  more  or  less  real  than  its  geometrical  shape.  We 
must  merely  learn  to  realize  that  the  shape  of  a  body  is  entirely  de- 
pendent on  the  particular  set  of  coordinates  chosen  for  the  making 
of  space  measurements. 

The  Kinematical  Rate  of  a  Clock. 

45.  Just  as  we  have  seen  that  the  shape  of  a  body  depends  upon 
our  choice  of  a  system  of  coordinates,  so  we  shall  find  that  the  rate  of 
a  given  clock  depends  upon  the  relative  motion  of  the  clock  and  its 
observer.  Consider  a  clock  or  any  mechanism  which  is  performing 
a  periodic  action.  Let  the  clock  be  at  rest  with  respect  to  system 
S'  and  let  a  given  period  commence  at  ti  and  end  at  t2f,  the  length  of 
the  interval  thus  being  A£'  =  t-2'  —  t\  . 

From  transformation  equation  (12)  we  may  obtain 


1--? 


and  by  subtraction,  since  x2  —  xi  is  obviously  equal  to  Vt,  we  have 

' 


1- 


At  =  —:=    =A<'. 


/        F2 

V1--? 


Kinemdtical  Applications.  51 

Thus  an  observer  who  is  moving  past  a  clock  finds  a  longer  period  for 

/        V* 
the  clock  in  the  ratio  1  :  \|  1  —  —  than  an  observer  who  is  stationary 

with  respect  to  it.  Suppose,  for  example,  we  have  a  particle  which 
is  turning  alternately  red  and  blue.  For  an  observer  who  is  moving 
past  the  particle  the  periods  for  which  it  remains  a  given  color  measure 

/        72 
longer  in  the  ratio  1  :  A/  1  —  —  than  they  do  to  an  observer  who  is 

stationary  with  respect  to  the  particle. 

46.  A  possible  opportunity  for  testing  this  interesting  conclusion 
of  the  theory  of  relativity  is  presented  by  the  phenomena  of  canal 
rays.     We  may  regard  the  atoms  which  are  moving  in  these  rays  as 
little  clocks,  the  frequency  of  the  light  which  they  emit  corresponding 
to  the  period  of  the  clock.     If  now  we  should  make  spectroscopic 
observations  on  canal  rays  of  high  velocity,  the  frequency  of  the 
emitted  light  ought  to  be  less  than  that  of  light  from  stationary  atoms 
of  the  same  kind  if  our  considerations  are  correct.     It  would  of  course 
be  necessary  to  view  the  canal  rays  at  right  angles  to  their  direction 
of  motion,  to  prevent  a  confusion  of  the  expected  shift  in  the  spectrum 
with  that  produced  by  the  ordinary  Doppler  effect  (see  Section  54). 

The  Idea  of  Simultaneity. 

47.  We  may  now  also  point  out  that  the  idea  of  the  absolute  simul- 
taneity of  two  events  must  henceforth  be  given  up.     Suppose,  for 
example,  an  observer  in  the  system  S  is  interested  in  two  events 
which  take  place  simultaneously  at  the  time  t.     Suppose  one  of  these 
events  occurs  at  a  point  having  the  X  coordinate  x\  and  the  other 
at  a  point  having  the  coordinate  xz;  then  by  transformation  equation 
(12)  it  is  evident  that  to  an  observer  in  system  S',  which  is  moving 
relative  to  S  with  the  velocity  V,  the  two  events  would  take  place 
respectively  at  the  times 


c2 
and 

"--HM -!••) 

V1  "* 


52    '  Chapter  Five. 

or  the  difference  in  time  between  the  occurrence  of  the  events  would 
appear  to  this  other  observer  to  be 

tj  -  t,r  = . —  (xi  -  x2).  (25) 


The  Composition  of  Velocities. 

48.  The  Case  of  Parallel  Velocities.  We  may  now  present  one  of 
the  most  important  characteristics  of  Einstein's  space  and  time, 
which  can  be  best  appreciated  by  considering  transformation  equation 
(14).  or  more  simply  its  analogue  for  the  transformation  in  the  reverse 
direction 


C2 


Consider  now  the  significance  of  the  above  equation.  If  we 
have  a  particle  which  is  moving  in  the  X  direction  with  the  velocity 
uxr  as  measured  in  system  £',  its  velocity  ux  with  respect  to  system  S 
is  to  be  obtained  by  adding  the  relative  velocity  of  the  two  systems  V 

ux'V 
and  dividing  the  sum  of  the  two  velocities  by  1  +  —  ^—  .     Thus  we  see 

that  we  must  completely  throw  overboard  our  old  naive  ideas  of  the 
direct  additivity  of  velocities.  Of  course,  in  the  case  of  very  slow 
velocities,  when  ux'  and  V  are  both  small  compared  with  the  velocity 

u/F 

of  light,  the  quantity  -^-  is  very  nearly  zero  and  the  direct  addition 

of  velocities  is  a  close  approximation  to  the  truth.  In  the  case  of 
velocities,  however,  which  are  in  the  neighborhood  of  the  speed  of 
light,  the  direct  addition  of  velocities  would  lead  to  extremely  er- 
roneous results. 

49.  In  particular  it  should  be  noticed  that  by  the  composition  of 
velocities  which  are  themselves  less  than  that  of  light  we  can  never 
obtain  any  velocity  greater  than  that  of  light.  Suppose,  for  example, 
that  the  system  Sf  were  moving  past  S  with  the  velocity  of  light 
(i.  e.,  V  =  c),  and  that  in  the  system  £'  a  particle  should  itself  be 
given  the  velocity  of  light  in  the  X  direction  (i.  e.,  uxf  —  c);  we  find 
on  substitution  that  the  particle  still  has  only  the  velocity  of  light 


Kinematical  Applications.  53 

with  respect  to  S.     We  have 

c  +  c       2c 

-  rrf  T  * 

If  the  relative  velocity  between  the  systems  should  be  one  half 

the  velocity  of  light,  -  ,  and  an  experimenter  on  S'  should  shoot  off  a 
z 

particle  in  the  X  direction  with  half  the  velocity  of  light,  the  total 
velocity  with  respect  to  S  would  be 


50.  Composition  of  Velocities  in  General.  In  the  case  of  particles 
which  have  components  of  velocity  in  other  than  the  X  direction  it 
is  obvious  that  our  transformation  equations  will  here  also  provide 
methods  of  calculation  to  supersede  the  simple  addition  of  velocities. 
If  we  place 


u'2  = 


we  may  obtain  by  the  substitution  of  equations  (14),  (15)  and  (16) 


#     .172.10    /T7  t^Psin^V/2 

+  V2  +  2u'F  cos  a  —  - 


u'Vcos* 


where  a  is  the  angle  in  the  system  S'  between  the  X'  axis  and  the 
velocity  of  the  particle  u'  .  For  the  particular  case  that  V  and  u' 
are  in  the  same  direction,  the  equation  obviously  reduces  to  the 
simpler  form 


which  we  have  already  considered. 

51.  We  may  also  call  attention  at  this  point  to  an  interesting  char- 
acteristic of  the  equations  for  the  transformation  of  velocities.     It  will 


54  Chapter  Five. 

be  noted  from  an  examination  of  these  equations  that  if  to  any  ob- 
server a  particle  appears  to  have  a  constant  velocity,  i.  e.,  to  be 
unacted  on  by  any  force,  it  will  also  appear  to  have  a  uniform  although 
of  course  different  velocity  to  any  observer  who  is  himself  in  uniform 
motion  with  respect  to  the  first.  An  examination,  however,  of  the 
transformation  equations  for  acceleration  (18),  (19),  (20)  will  show 
that  here  a  different  state  of  affairs  is  true,  since  it  will  be  seen  that  a 
point  which  has  uniform  acceleration  (x,  y,  z)  with  respect  to  an  ob- 
server in  system  S  will  not  in  general  have  a  uniform  acceleration  in 
another  system  S',  since  the  acceleration  in  system  Sf  depends  not 
only  on  the  constant  acceleration  but  also  on  the  velocity  in  system  S, 
which  is  necessarily  varying. 

Velocities  Greater  than  that  of  Light. 

52.  In  the  preceding  section  we  have  called  attention  to  the  fact 
that  the  mere  composition  of  velocities  which  are  not  themselves 
greater  than  that  of  light  will  never  lead  to  a  speed  that  is  greater 
than  that  of  light.  The  question  naturally  arises  whether  velocities 
which  are  greater  than  that  of  light  could  ever  possibly  be  obtained 
in  any  way. 

This  problem  can  be  attacked  in  an  extremely  interesting  manner. 
Consider  two  points  A  and  B  on  the  X  axis  of  the  system  S,  and 
suppose  that  some  impulse  originates  at  A,  travels  to  B  with  the 
velocity  u  and  at  B  produces  some  observable  phenomenon,  the  start- 
ing of  the  impulse  at  A  and  the  resulting  phenomenon  at  B  thus 
being  connected  by  the  relation  of  cause  and  effect. 

The  time  elapsing  between  the  cause  and  its  effect  as  measured 
in  the  units  of  system  S  will  evidently  be 

H-b-tA-**-,  (28) 


where  XA  and  XB  are  the  coordinates  of  the  two  points  A  and  B. 

Now  in  another  system  S',  which  has  the  velocity  V  with  respect 
to  S,  the  time  elapsing  between  cause  and  effect  would  evidently  be 

*-4-£-       l      l        V    ^          1      /        V 


Kinemdtical  Applications.  55 

where  we  have  substituted  for  t'B  and  l'A  in  accordance  with  equation 
(12).     Simplifying  and  introducing  equation  (28)  we  obtain 

I-*? 

-A*.  (29) 


V2 


Let  us  suppose  now  that  there  are  no  limits  to  the  possible  magni- 
tude of  the  velocities  u  and  7,  and  in  particular  that  the  causal  im- 
pulse can  travel  from  A  to  B  with  a  velocity  u  greater  than  that  of 
light.  It  is  evident  that  we  could  then  take  a  velocity  u  great  enough 

uV 

so  that  —  would  be  greater  than  unity  and  A2'  would  become  nega- 
tive. In  other  words,  for  an  observer  in  system  S'  the  effect  which 
occurs  at  B  would  precede  in  time  its  cause  which  originates  at  A. 
Such  a  condition  of  affairs  might  not  be  a  logical  impossibility;  never- 
theless its  extraordinary  nature  might  incline  us  to  believe  that  no 
causal  impulse  can  travel  with  a  velocity  greater  than  that  of  light. 
We  may  point  out  in  passing,  however,  that  in  the  case  of  kine- 
matic occurrences  in  which  there  is  no  causal  connection  there  is  no 
reason  for  supposing  that  the  velocity  must  be  less  than  that  of  light. 
Consider,  for  example,  a  set  of  blocks  arranged  side  by  side  in  a  long 
row.  For  each  block  there  could  be  an  independent  time  mechanism 
like  an  alarm  clock  which  would  go  off  at  just  the  right  instant  so 
that  the  blocks  would  fall  down  one  after  another  along  the  line. 
The  velocity  with  which  the  phenomenon  would  travel  along  the 
line  of  blocks  could  be  arranged  to  have  any  value.  In  fact,  the 
blocks  could  evidently  all  be  fixed  to  fall  just  at  the  same  instant, 
which  would  correspond  to  an  infinite  velocity.  It  is  to  be  noticed 
here,  however,  that  there  is  no  causal  connection  between  the  falling 
of  one  block  and  that  of  the  next,  and  no  transfer  of  energy. 

Application  of  the  Principles  of  Kinematics  to  Certain  Optical  Prob- 
lems. 

53.  Let  us  now  apply  our  kinematical  considerations  to  some 
problems  in  the  field  of  optics.  We  may  consider  a  beam  of  light 
as  a  periodic  electromagnetic  disturbance  which  is  propagated  through 
a  vacuum  with  the  velocity  c.  At  any  point  in  the  path  of  a  beam  of 


56  Chapter  Five. 

light  the  intensitjr  of  the  electric  and  magnetic  fields  will  be  undergoing 
periodic  changes  in  magnitude.  Since  the  intensities  of  both  the 
electric  and  the  magnetic  fields  vary  together,  the  statement  of  a 
single  vector  is  sufficient  to  determine  the  instantaneous  condition 
at  any  point  in  the  path  of  a  beam  of  light.  It  is  customary  to  call 
this  vector  (which  might  be  either  the  strength  of  the  electric  or  of 
the  magnetic  field)  the  light  vector. 

For  the  case  of  a  simple  plane  wave  (i.  e.,  a  beam  of  monochromatic 
light  from  a  distant  source)  the  light  vector  at  any  point  in  the  path 
of  the  light  may  be  put  proportional  to 


te  +  7  +  "2).  (30) 


where  x,  y  and  z  are  the  coordinates  of  the  point  under  observation, 
t  is  the  time,  Z,  m  and  n  are  the  cosines  of  the  angles  a,  /3  and  7  which 
determine  the  direction  of  the  beam  of  light  with  reference  to  our 
system,  and  co  is  a  constant  which  determines  the  period  of  the  light. 
If  now  this  same  beam  of  light  were  examined  by  an  observer  in 
system  S'  which  is  moving  past  the  original  system  in  the  X  direction 
with  the  velocity  V,  we  could  write  the  light  vector  proportional  to 


(31) 


It  is  not  difficult  to  show  that  the  transformation  equations  which 
we  have  already  developed  must  lead  to  the  following  relations  between 
the  measurements  in  the  two  systems* 

*  Methods  for  deriving  the  relation  between  the  accented  and  unaccented 
quantities  will  be  obvious  to  the  reader.  For  example,  consider  the  relation  between 
co  and  co'.  At  the  origin  of  coordinates  x  =  y  =  z  =  Qin  system  S,  we  shall  have 
in  accordance  with  expression  (30)  the  light  vector  proportional  to  sin  ut,  and  hence 
similarly  at  the  point  0',  which  is  the  origin  of  coordinates  in  system  S',  we  shall 
have  the  light  vector  proportional  to  sin  coT.  But  the  point  0'  as  observed  from 
system  S  moves  with  the  velocity  V  along  the  X-axis  and  at  any  instant  has  the 
position  x  =  Vt'}  hence  substituting  in  expression  (30)  we  have  the  light  vector  at 
the  point  0'  as  measured  in  system  S  proportional  to 

sin  co*  I  1  —  I  —  J  ,  (36) 

while  as  measured  in  system  S'  the  intensity  is  proportional  to 

sin  coT.  (37) 


Kinematical  Applications.  57 

(32) 


l-l 

4'  (33) 

1  -I- 

c 


K  \    1  - 

n 


(34) 
(35) 


With  the  help  of  these  equations  we  may  now  treat  some  important 
optical  problems. 

54.  The  Doppler  Effect.  At  the  origin  of  coordinates,  x  =  y  =  z 
=  0,  in  system  S  we  shall  evidently  have  from  expression  (30)  the 
light  vector  proportional  to  sin  ut.  That  means  that  the  vector 
becomes  zero  whenever  cot  =  2Nir,  where  N  is  any  integer;  in  other 

2w 
words,  the  period  of  the  light  is  p  =  -  -  or  the  frequency 

CO 


27T* 

Similarly  the  frequency  of  the  light  as  measured  by  an  observer  in 
system  S'  would  be 


We  have  already  obtained,  however,  a  transformation  equation  for  t',  namely, 


and  further  may  place  x  =  Vt.     Making  these  substitutions  and  comparing  ex- 
pressions (36)  and  (37)  we  see  that  we  must  have  the  relation 


CO      =    COK   1     1    —   t  —    I   . 


Methods  of  obtaining  the  relation  between  the  cosines  I,  m  and  n  and  the  corre- 
sponding cosines  I',  m',  and  n'  as  measured  in  system  S'  may  be  left  to  the  reader. 


58  Chapter  Five. 

Combining  these  two  equations  and  substituting  the  equation  con- 
necting co  and  co'  we  have 


v  = 


K  i  i  —  i  — 


This  is  the  relation  between  the  frequencies  of  a  given  beam  of  light 
as  it  appears  to  observers  who  are  in  relative  motion. 

If  we  consider  a  source  of  light  at  rest  with  respect  to  system  S' 
and  at  a  considerable  distance  from  the  observer  in  system  St  we 
may  substitute  for  v'  the  frequency  of  the  source  itself,  VQ}  and  for  I 
we  may  write  cos  </>,  where  <f>  is  the  angle  between  the  line  connecting 
source  and  observer  and  the  direction  of  motion  of  the  source,  leading 
to  the  expression 


(38) 


This  is  the  most  general  equation  for  the  Doppler  effect.  When 
the  source  of  light  is  moving  directly  in  the  line  connecting  source 
and  observer,  we  have  cos  <£  =  1,  and  the  equation  reduces  to 

(39) 


V\ 

1 

C   J 


which  except  for  second  order  terms  is  identical  with  the  older  ex- 
pressions for  the  Doppler  effect,  and  hence  agrees  with  experimental 
determinations. 

We  must  also  observe,  however,  that  even  when  the  source  of 
light  moves  at  right  angles  to  the  line  connecting  source  and  observer 
there  still  remains  a  second-order  effect  on  the  observed  frequency, 
in  contradiction  to  the  predictions  of  older  theories.  We  have  in  this 
case  cos  0  =  0, 


V    — 


(40) 


This  is  the  change  in  frequency  which  we  have  already  considered 
when  we  discussed  the  rate  of  a  moving  clock.     The  possibilities  of 


Kinematical  Applications.  59 

direct  experimental  verification  should  not  be  overlooked  (see  sec- 
tion 46). 

55.  The  Aberration  of  Light.  Returning  now  to  our  transforma- 
tion equations,  we  see  that  equation  (33)  provides  an  expression  for 
calculating  the  aberration  of  light.  Let  us  consider  that  the  source 
of  light  is  stationary  with  respect  to  system  S,  and  let  there  be  an 
observer  situated  at  the  origin  of  coordinates  of  system  S'  and  thus 
moving  past  the  source  with  the  velocity  V  in  the  X  direction.  Let  <j> 
be  the  angle  between  the  X-axis  and  the  line  connecting  source  of 
light  and  observer  and  let  <£'  be  the  same  angle  as  it  appears  to  the 
moving  observer;  then  we  can  obviously  substitute  in  equation  (33), 
cos  <£  =  I,  cos  <t>'  =  I',  giving  us 

F 

cos  <j>  —  — 

cos  </>'  =  -  -^.  (41) 

1  —  cos  <f>  — 
c 

This  is  a  general  equation  for  the  aberration  of  light. 

For  the  particular  case  that  the  direction  of  the  beam  of  light  is 
perpendicular  to  the  motion  of  the  observer  we  have  cos  <f>  =  0 

V 

cos  <£'  =  -  — ,  (42) 

C 

which,  except  for  second-ordeii  differences,  is  identical  with  the  familiar 
expression  which  makes  the  tangent  of  the  angle  of  aberration  nu- 
merically equal  to  V/c.  The  experimental  verification  of  the  formula 
by  astronomical  measurements  is  familiar. 

56.  Velocity  of  Light  in  Moving  Media.  It  is  also  possible  to  treat 
very  simply  by  kinematic  methods  the  problem  of  the  velocity  of 
light  in  moving  media.  We  shall  confine  ourselves  to  the  particular 
case  of  a  beam  of  light  in  a  medium  which  is  itself  moving  parallel 
to  the  light. 

Let  the  medium  be  moving  with  the  velocity  V  in  the  X  direction, 
and  let  us  consider  the  system  of  coordinates  S'  as  stationary  with 
respect  to  the  medium.  Now  since  the  medium  appears  to  be  sta- 
tionary with  respect  to  observers  in  S'  it  is  evident  that  the  velocity 
of  the  light  with  respect  to  S'  will  be  c/n,  where  n  is  index  of  refraction 


60  Chapter  Five. 

for  the  medium.  If  now  we  use  our  equation  (26)  for  the  addition  of 
velocities  we  shall  obtain  for  the  velocity  of  light,  as  measured  by 
observers  in  S, 


(43) 


Carrying  out  the  division  and  neglecting  terms  of  higher  order  we 
obtain 

(44) 

The  equation  thus  obtained  is  identical  with  that  of  Fresnel,  the 

/  M2  -  1  \ 
quantity  ( —  J  being  the  well-known  Fresnel  coefficient.     The 

empirical  verification  of  this  equation  by  the  experiments  of  Fizeau 
and  of  Michelson  and  Morley  is  too  well  known  to  need'  further 
mention. 

For  the  case  of  a  dispersive  medium  we  should  obviously  have  to 
substitute  in  equation  (44)  the  value  of  /*  corresponding  to  the  par- 
ticular frequency,  /,  which  the  light  has  in  system  S'.  It  should  be 
noticed  in  this  connection  that  the  frequencies  /  and  v  which  the 
light  has  respectively  in  system  S  and  system  S',  although  nearly 
enough  the  same  for  the  practical  use  of  equation  (44),  are  in  reality 
connected  by  an  expression  which  can  easily  be  shown  (see  section  54) 
to  have  the  form 

/         v\ 

(45) 

57.  Group  Velocity.  In  an  entirely  similar  way  we  may  treat  the 
problem  of  group  velocity  and  obtain  the  equation 

G'  +  V 
G  =  ~   -7^T>  (46) 


where  G'  is  the  group  velocity  as  it  appears  to  an  observer  who  is 


Kinematical  Applications.  61 

stationary  with  respect  to  the  medium.  G'  is,  of  course,  an  experi- 
mental quantity,  connected  with  frequency  and  the  properties  of  the 
medium,  in  a  way  to  be  determined  by  experiments  on  the  stationary 
medium. 

In  conclusion  we  wish  to  call  particular  attention  to  the  extra- 
ordinary simplicity  of  this  method  of  handling  the  optics  of  moving 
media  as  compared  with  those  that  had  to  be  employed  before  the 
introduction  of  the  principle  of  relativity. 


CHAPTER  VI. 
THE  DYNAMICS  OF  A  PARTICLE. 

58.  In  this  chapter  and  the  two  following,  we  shall  present  a 
system  of  "  relativity  mechanics  "  based  on  Newton's  three  laws  of 
motion,  the  Einstein  transformation  equations  for  space  and  time, 
and  the  principle  of  the  conservation  of  mass. 

The  Laws  of  Motion. 

Newton's  laws  of  motion  may  be  stated  in  the  following  form: 

I.  Every  particle  continues  in  its  state  of  rest  or  of  uniform  motion 
in  a  straight  line,  unless  it  is  acted  upon  by  an  external  force. 

II.  The  rate  of  change  of  the  momentum  of  the  particle  is  equal 
to  the  force  acting  and  is  in  the  same  direction. 

III.  For  the  action  of  every  force  there  is  an  equal  force  acting 
in  the  opposite  direction. 

Of  these  laws  the  first  two  merely  serve  to  define  the  concept  of 
force,  and  their  content  may  be  expressed  in  mathematical  form  by 
the  following  equation  of  definition 

•I t>  e £  *0  t**A C    OtfFfRlc^cf 

a**  ?•**&  A»O  *tt*L€f#47i9*i  T*      d  du      dm 

:«ft  «-  *er^.  F  =  dt  (mU)  =  m  Tt   +  #"  U'  (47) 

where  F  is  the  force  acting  on  a  particle  of  mass  m  which  has  the 
velocity  u,  and  hence  the  momentum  rau. 

Quite  different  in  its  nature  from  the  first  two  laws,  which  merely 
give  us  a  definition  of  force,  the  third  law  states  a  very  definite  physical 
postulate,  since  it  requires  for  every  change  in  the  momentum  of  a 
body  an  equal  and  opposite  change  in  the  momentum  of  some  other 
body.  The  truth  of  this  postulate  will  of  course  be  tested  by  com- 
paring with  experiment  the  results  of  the  theory  of  mechanics  which 
we  base  upon  its  assumption. 

Difference  between  Newtonian  and  Relativity  Mechanics. 

59.  Before  proceeding  we  may  point  out  the  particular  difference 
between  the  older  Newtonian  mechanics,  which  were  based  on  the 
laws  of  motion  and  the  Galilean  transformation  equations  for  space 

62 


Dynaniics  of  a  Particle.  63 

and  time,  and  our  new  system  of  relativity  mechanics  based  on 
those  same  laws  of  motion  and  the  Einstein  transformation  equations. 
In  the  older  mechanics  there  was  no  reason  for  supposing  that  the 
mass  of  a  body  varied  in  any  way  with  its  velocity,  and  hence  force 
could  be  defined  interchangeably  as  the  rate  of  change  of  momentum 
or  as  mass  times  acceleration,  since  the  two  were  identical.  In  rela- 
tivity mechanics,  however,  we  shall  be  forced  to  conclude  that  the 
mass  of  a  body  increases  in  a  perfectly  definite  way  with  its  velocity, 
and  hence  in  our  new  mechanics  we  must  define  force  as  equal  to  the 
total  rate  of  change  of  momentum  t 

d(mu)          du.      dm 

IT  =mdi+~dt* 

dn 

instead  of  merely  as  mass  times  acceleration  m  -77  .     If  we  should  try 

cit 

to  define  force  in  "  relativity  mechanics  "  as  merely  equal  to  mass 
times  acceleration,  we  should  find  that  the  application  of  Newton's 
third  law  of  motion  would  then  lead  to  very  peculiar  results,  which 
would  make  the  mass  of  a  body  different  in  different  directions  and 
force  us  to  give  up  the  idea  of  the  conservation  of  mass. 

The  Mass  of  a  Moving  Particle. 

60.  In  Section  31  we  have  already  obtained  in  an  elementary  way 
an  expression  for  the  mass  of  a  moving  particle,  by  considering  a 
collision  between  elastic  particles  and  calculating  how  the  resulting 
changes  in  velocity  would  appear  to  different  observers   who   are 
themselves  in  relative  motion.     Since  we  now  have  at  our  command 
general  formulae  for  the  transformation  of  velocities,  we  are  now  in 
a  position  to  handle  this  problem  much  more  generally,  and  in  particu- 
lar to  show  that  the  expression  obtained  for  the  mass  of  a  moving  particle 
is  entirely  independent  of  the  consideration  of  any  particular  type  of 
collision. 

61.  Transverse  Collision.     Let  us  first  treat  the  case  of  a  so-called 
"  transverse  "  collision.     Consider  a  system  of  coordinates  and  two 
exactly  similar  elastic  particles,  each  having  the  mass  mo  when  at 
rest,  one  moving  in  the  X  direction  with  the  velocity  -f-  u  and  the 
other  with  the  velocity   —  u.     (See  figure   11.)     Besides  the  large 
components  of  velocity  +  u  and  —  u  which  they  have  in  the  X  direc- 


64  Chapter  Six. 

tion  let  them  also  have  small  components  of  velocity  in  the  Y  direc- 
tion, +  v  and  —  v.  The  experiment  is  so  arranged  that  the  particles 
will  just  undergo  a  glancing  collision  as  they  pass  each  other  and 

rebound  with   components 

_v  of  velocity  in  the  Y  direc- 

Y  +v  tion  of  the  same  magnitude, 

C          -u  z;,  which  they  originally  had, 

but  in  the  reverse  direction. 

(It  is  evident  from  the  symmetry  of  the  arrangement  that  the  experi- 
ment would  actually  occur  as  we  have  stated.) 

We  shall  now  be  interested  in  the  way  this  experiment  would  appear 
to  an  observer  who  is  in  motion  in  the  X  direction  with  the  velocity  V 
relative  to  our  original  system  of  coordinates. 

From  equation  (14)  for  the  transformation  of  velocities,  it  can 
be  seen  that  this  new  observer  would  find  for  the  X  component  velocities 
of  the  two  particles  the  values 

u-V  -u-V 

Ul  =       ~uV        and        U2  =         ~uV~ 


and  from  equation  (15)  for  the  Y  component  velocities  would  find  the 
values 


(49) 


c2 


the  signs  depending  on  whether  the  velocities  are  measured  before  or 
after  the  collision. 

Now  from  Newton's  third  law  of  motion  (i.  e.,  the  principle  of 
the  equality  of  action  and  reaction)  it  is  evident  that  on  collision 
the  two  particles  must  undergo  the  same  numerical  change  in  momen- 
tum. 

For  the  experiment  that  we  have  chosen  the  only  change  in  mo- 
mentum is  in  the  Y  direction,  and  the  observer  whose  measurements 
we  are  considering  finds  that  one  particle  undergoes  the  total  change 


Dynamics  of  a  Particle. 


65 


in  velocity 


V2 


20!    = 


'  c* 


and  that  the  other  particle  undergoes  the  change  in  velocity 


Since  these  changes  in  the  velocities  of  the  particles  are  not  equal, 
it  is  evident  that  their  masses  must  also  be  unequal  if  the  principle 
of  the  equality  of  action  and  reaction  is  true  for  all  observers,  as  we 
have  assumed.  This  difference  in  the  mass  of  the  particles,  each  of 
which  has  the  mass  ra0  when  at  rest,  arises  from  the  fact  that  the  mass 
of  a  particle  is  a  function  of  its  velocity  and  for  the  observer  in  question 
the  two  particles  are  not  moving  with  the  same  velocity. 

Using  the  symbols  mi  and  w2  for  the  masses  of  the  particles,  we 
may  now  write  as  a  mathematical  expression  of  the  requirements  of 
the  third  law  of  motion 


V2 

1  -  -         2m2v 


,  uV  uV 

c2  c2 

Simplifying,  we  obtain  by  direct  algebraic  transformation 


uV 


mi 
nit 


N 

!       /- 

w-  yy 

(' 

+f) 

c2 

\ 

/  «  -  F  V 

,lj 

w7  1 
c2  1 

c2 

66  Chapter  Six. 

which  on  the  substitution  of  equations  (48)  gives  us 


=  •  (50) 

<? 

This  equation  thus  shows  that  the  mass  of  a  particle  moving  with 

/         u2 
the  velocity  u*  is  inversely  proportional  to  -^1  —  -- ,  and,  denoting 

the  mass  of  the  particle  at  rest  by  mQ,  we  may  write  as  a  general  ex- 
pression for  the  mass  of  a  moving  particle 

/m  _ 

(51) 


62.  Mass  the  Same  in  All  Directions.  The  method  of  derivation 
that  we  have  just  used  to  obtain  this  expression  for  the  mass  of  a 
moving  particle  is  based  on  the  consideration  of  a  so-called  "  trans- 
verse collision,"  and  in  fact  the  expression  obtained  has  often  been 
spoken  of  as  that  for  the  transverse  mass  of  a  moving  particle,  while 

a  different  expression,  —     ^  ^3/2,  has  been  used  for  the  so-called 


\  X 

longitudinal  mass  of  the  particle.     These  expressions  — 7=       =  and 


('-£)•" 


are,  as  a  matter  of  fact,  the  values  of  the  electric  force 


necessary  to  give  a  charged  particle  unit  acceleration  respectively 
at  right  angles  and  in  the  same  direction  as  its  original  velocity,  and 
hence  such  expressions  would  be  proper  for  the  mass  of  a  moving  par- 
ticle if  we  should  define  force  as  mass  times  acceleration.  As  already 

*  For  simplicity  of  calculation  we  consider  the  case  where  the  components  of 
velocity  in  the  Y  direction  are  small  enough  to  be  negligible  in  their  effect  on  the 
mass  of  the  particles  compared  with  the  large  components  of  velocity  u\  and  u2  in 
the  X  direction. 


Dynarftics  of  a  Particle.  67 

stated,  however,  it  has  seemed  preferable  to  retain,  for  force,  Newton's 
original  definition  which  makes  it  equal  to  the  rate  of  change  of 
momentum,  and  we  shall  presently  see  that  this  more  suitable  defini- 
tion is  in  perfect  accord  with  the  idea  that  the  mass  of  a  particle  is 
the  same  in  all  directions. 

Aside  from  the  unnecessary  complexity  which  would  be  intro- 
duced, the  particular  reason  making  it  unfortunate  to  have  different 
expressions  for  mass  in  different  directions  is  that  under  such  con- 
ditions it  would  be  impossible  to  retain  or  interpret  the  principle  of 
the  conservation  of  mass.  And  we  shall  now  proceed  to  show  that 
by  introducing  the  principle  of  the  conservation  of  mass,  the  con- 
sideration of  a  "  longitudinal  collision  "  will  also  lead  to  exactly  the 

same  expression,  —  j=  =  ,  for  the  mass  of  a  moving  particle  as  we 


have  already  obtained  from  the  consideration  of  a  transverse  collision. 

63.  Longitudinal  Collision.  Consider  a  system  of  coordinates  and 
two  elastic  particles  moving  in  the  X  direction  with  the  velocities 
+  u  and  —  u  so  that  a  "  longitudinal  "  (i.  e.,  head-on)  collision  will 
occur.  Let  the  particles  be  exactly  alike,  each  of  them  having  the 
mass  mQ  when  at  rest.  On  collision  the  particles  will  evidently  come 
to  rest,  and  then  under  the  action  of  the  elastic  forces  developed  start 
up  and  move  back  over  their  original  paths  with  the  respective  veloci- 
ties —  u  and  +  u  of  the  same  magnitude  as  before. 

Let  us  now  consider  how  this  collision  would  appear  to  an  observer 
who  is  moving  past  the  original  system  of  coordinates  with  the  velocity 
V  in  the  X  direction.  Let  HI  and  u*  be  the  velocities  of  the  particles 
as  they  appear  to  this  new  observer  before  the  collision  has  taken 
place.  Then,  from  our  formula  for  the  transformation  of  velocities 
(14),  it  is  evident  that  we  shall  have 


c2  l+   c2 

Since  these  velocities  u\  and  u2  are  not  of  the  same  magnitude, 
the  two  particles  which  have  the  same  mass  when  at  rest  do  not  have 
the  same  mass  for  this  observer.  Let  us  call  the  masses  before  col- 
lision mi  and  w2. 


68  Chapter  Six. 

Now  during  the  collision  the  velocities  of  the  particles  will  all  the 
time  be  changing,  but  from  the  principle  of  the  conservation  of  mass 
the  sum  of  the  two  masses  must  all  the  time  be  equal  to  mi  +  m2. 
When  in  the  course  of  the  collision  the  particles  have  come  to  relative 
rest,  they  will  be  moving  past  our  observer  with  the  velocity  —  V, 
and  their  momentum  will  be  —  (mi  +  m2)F.  But,  from  the  principle 
of  the  equality  of  action  and  reaction,  it  is  evident  that  this  momen- 
tum must  be  equal  to  the  original  momentum  before  collision  occurred. 
This  gives  us  the  equation  —  (mi  +  m2)F  =  m^i  +  m2^2.  Substi- 
tuting our  values  (52)  for  u\  and  uz  we  have 

mi  w2 


and  by  direct  algebraic  transformation,   as  in  the  previous  proof, 
this  can  be  shown  to  be  identical  with 


mi 


/  Ui^ 

V1 "  7" 


leading  to  the  same  expression  that  we  obtained  before  for  the  mass 
of  a  moving  particle,  viz.: 

m0 


m  = 


64.  Collision  of  Any  Type.  We  have  derived  this  formula  for  the 
mass  of  a  moving  particle  first  from  the  consideration  of  a  transverse 
and  then  of  a  longitudinal  collision  between  particles  which  are  elastic 
and  have  the  same  mass  when  at  rest.  It  seems  to  be  desirable  to 
show,  however,  that  the  consideration  of  any  type  of  collision  between 
particles  of  any  mass  leads  to  the  same  formula  for  the  mass  of  a 
moving  particle. 

For  the  mass  m  of  a  particle  moving  with  the  velocity  u  let  us 
write  the  equation  m  =  m0F(w2),  where  F(  )  is  the  function  whose 
form  we  wish  to  determine.  The  mass  is  written  as  a  function  of 


Dynamics  of  a  Particle.  69 

the  square  of  the  velocity,  since  from  the  homogeneity  of  space  the 
mass  will  be  independent  of  the  direction  of  the  velocity,  and  the 
mass  is  made  proportional  to  the  mass  at  rest,  since  a  moving  body 
may  evidently  be  thought  of  as  divided  into  parts  without  change  in 
mass.  It  may  be  further  remarked  that  the  form  of  the  function 
F(  )  must  be  such  that  its  value  approaches  unity  as  the  variable 
approaches  zero. 

Let  us  now  consider  two  particles  having  respectively  the  masses 
mo  and  n0  when  at  rest,  moving  with  the  velocities  u  and  w  before 
collision,  and  with  the  velocities  U  and  W  after  a  collision  has  taken 
place. 

From  the  principle  of  the  conservation  of  mass  we  have 

moF(ux2  +  uv2  +  u?)  +  nQF(wx2  +  wf  +  w*) 

=  mQF(Ux2  +  Uv*  +  U2)  +  n,F(W2  +  Wv2  +  W2},     (53) 

and  from  the  principle  of  the  equality  of  action  and  reaction  (i.  e., 
Newton's  third  law  of  motion) 

mQF(ux2  +  uy2  +  u2)ux  +  noF(ti>-2  +  w2  +  w2)wx 

=  moF(Ux2  +  Uu2  +  U2)UX  +  n,F(W2  +  Wy2  +  W2)W2,     (54) 
mQF(ux2  +  u2  +  w.2K  +  nQF(wx2  -f  wy2  +  w2)wv 

=  m*F(U2  +  U2  +  U2}Uy  +  n,F(W2  +  W2  +  W2)Wy,     (55) 
mQF(ux2  +  u2  +  u?)uz  +  n<>F(w2  +  w2  +  w2)w, 

+  U2  -f  U2)UZ  +  n,F(W2  +  Wy2  +  W2)WZ.     (56) 


These  velocities,  uX)  uv,  uz)  wx,  wv,  wf,  Ux,  etc.,  are  measured,  of 
course,  with  respect  to  some  definite  system  of  "  space-time  "  coordi- 
nates. An  observer  moving  past  this  system  of  coordinates  with  the 
velocity  V  in  the  X  direction  would  find  for  the  corresponding  com- 
ponent velocities  the  values 


r      /,   YI 

'*  ~* 


U 


u^V'  u^V"  u,V"  w.V'         '' 

*  (f  #  <? 

as  given  by  our  transformation  equations  for  velocity  (14,  15,  16). 


70 


Chapter  Six. 


Since  the  law  of  the  conservation  of  mass  and  Newton's  third 
law  of  motion  must  also  hold  for  the  measurements  of  the  new  ob- 
server; we  may  write  the  following  'new  relations  corresponding  to 
equations  53  to  56: 


m0F 


ux-V 


1  - 


mQF{ui 


c2 


i  


uxV 


UxV 


(53a) 


-  V 


1  - 


ux-v 


I  - 


W    —  V 


I  -  - 


(54a) 


72 


72 

c2 


1  - 


wfV 


72 


=  mQF{Ux 


(55a) 


u, 


Dynamics  of  a  Particle.  71 

V*  I        V* 

. 

+n0F[wf-- 


~  ~      C2 

i  -  (56a) 


1  —        o  1 


It  is  evident  that  these  equations  (53a-56a)  must  be  true  no 
matter  what  the  velocity  between  the  new  observer  and  the  original 
system  of  coordinates,  that  is,  true  for  all  values  of  V.  The  velocities 
ux,  uv,  uz,  wx,  etc.,  are,  however,  perfectly  definite  quantities,  measured 
with  reference  to  a  definite  system  of  coordinates  and  entirely  inde- 
pendent of  V.  If  these  equations  are  to  be  true  for  perfectly  definite 
values  of  ux,  uy,  uz,  wxi  etc.,  and  for  all  values  of  V,  it  is  evident  that 
the  function  F(  )  must  be  of  such  a  form  that  the  equations  are 
identities  in  V.  As  a  matter  of  fact,  it  is  found  by  trial  that  V  can 
be  cancelled  from  all  the  equations  if  we  make  F(  )  of  the  form 

;  and  we  see  that  the  expected  relation  is  a  solution  of  the 


equations,  although  perhaps  not  necessarily  a  unique  solution. 

Before  proceeding  to  use  our  formula  for  the  mass  of  a  moving 
particle  for  the  further  development  of  our  system  of  mechanics, 
we  may  call  attention  in  passing  to  the  fact  that  the  experiments  of 
Kaufmann,  Bucherer,  and  Hupka  have  in  reality  shown  that  the  mass 
of  the  electron  increases  with  its  velocity  according  to  the  formula 
which  we  have  just  obtained.  We  shall  consider  the  dynamics  of  the 
electron  more  in  detail  in  the  chapter  devoted  to  electromagnetic 
theory.  We  wish  to  point  out  now,  however,  that  in  this  derivation 
we  have  made  no  reference  to  any  electrical  charge  which  might  be 
carried  by  the  particle  whose  mass  is  to  be  determined.  Hence  we 
may  reject  the  possibility  of  explaining  the  Kaufmann  experiment 
by  assuming  that  the  charge  of  the  electron  decreases  with  its  velocity, 
since  the  increase  in  mass  is  alone  sufficient  to  account  for  the  results 
of  the  measurement. 


72  Chapter  Six. 

Transformation  Equations  for  Mass. 

65.  Since  the  velocity  of  a  particle  depends  on  the  particular 
system  of  coordinates  chosen  for  the  measurement,  it  is  evident  that 
the  mass  of  the  particle  will  also  depend  on  our  reference  system  of 
coordinates.  For  the  further  development  of  our  system  of  dynamics, 
we  shall  find  it  desirable  to  obtain  transformation  equations  for  mass 
similar  to  those  already  obtained  for  velocity,  acceleration,  etc. 

We  have 


m  = 


tf 

-L T 


where  the  velocity  u  is  measured  with  respect  to  some  definite  system 
of  coordinates,  S.  Similarly  with  respect  to  a  system  of  coordinates 
S'  which  is  moving  relatively  to  S  with  the  velocity  V  in  the  X  direc- 
tion we  shall  have 


We  have  already  obtained,  however,  a  transformation  equation  (17) 
for  the  function  of  the  velocity  occurring  in  these  equations  and  on 
substitution  we  obtain  the  desired  transformation  equation 


(57) 


where  K  has  the  customary  significance 

1 


By  differentiation  of  (57)  with  respect  to  the  time  and  simpli- 
fication, we  obtain  the  following  transformation  equation  for  the 
rate  at  which  the  mass  of  a  particle  is  changing  owing  to  change  in 
velocity 

xV\-ldux 

-'•  (58) 


Dynamics  of  a  Particle. 


73 


Equation  for  the  Force  Acting  on  a  Moving  Particle. 

66.  We  are  now  in  a  position  to  return  to  our  development  of  the 
dynamics  of  a  particle.  In  the  first  place,  the  equation  which  we 
have  now  obtained  for  the  mass  of  a  moving  particle  will  permit 
us  to  rewrite  the  original  equation  by  which  we  defined  force,  in  a 
number  of  ways  which  will  be  useful  for  future  reference. 

We  have  our  equation  of  definition  (47) 


d 


d\i      dm 


which,  on  substitution  of  the  expression  for  m,  gives  us 


mo 


_du       d_ 
'~tfdt+dt 


or,  carrying  out  the  indicated  differentiation, 


F  = 


mo 


<? 


u     (59) 


(60) 


Transformation  Equations  for  Force. 

67.  We  are  also  in  position  to  obtain  transformation  equations  for 
force.     We  have 


F  =  —  (rau)  =  mu 


rau 


or 


mux, 


Fx  —  mux 
Fv  —  muv 
Fz  =  muz 

We  have  transformation  equations,  however,  for  all  the  quantities 
on  the  right-hand  side  of  these  equations.  For  the  velocities  we 
have  equations  (14),  (15)  and  (16),  for  the  accelerations  (18),  (19) 
and  (20),  for  mass,  equation  (57)  and  for  rate  of  change  of  mass, 
equation  (58).  Substituting  above  we  obtain  as  our  transformation 


74 

equations  for  force 


Chapter  Six. 


Fx  -  mV 


1  - 


uxV 


Fx- 


UyV 


1  - 


uzV 


(62) 


IV- 


1  - 


uxV 


(63) 


We  may  now  consider  a  few  applications  of  the  principles  governing 
the  dynamics  of  a  particle. 

The  Relation  between  Force  and  Acceleration. 

68.  If  we  examine  our  equation   (59). for  the  force  acting  on  a 
particle  « 

/w»  _  x7«-f  s7  /wi  _ 

i,  (59) 


we  see  that  the  force  is  equal  to  the  sum  of  two  vectors,  one  of  which 

du 

is  in  the  direction  of  the  acceleration  -7-  and  the  other  in  the  direction 

at 

of  the  existing  velocity  u,  so  that  in  general  force  and  the  acceleration 

it  produces  are  not  in  the  same  di- 
rection. We  shall  find  it  interesting 
to  see,  however,  that  if  the  force 
which  does  produce  acceleration  in 
a  given  direction  be  resolved  per- 
pendicular and  parallel  to  the  accel- 
eration, the  two  components  will 
be  connected  by  a  definite  relation. 
Consider  a  particle  (fig.  12)  in 
plane  space  moving  with  the  ve- 
locity 

O 

FIG.  12.  u  =  uxi  +  uyj. 


Dynamics  of  a  Particle. 


75 


Let  it  be  accelerated  in  the  X  direction  by  the  action  of  the  com- 
ponent forces  Fx  and  Fv. 

From  our  general  equation  (59)  for  the  force  acting  on  a  particle 
we  have  for  these  component  forces 


Fx  = 


F    = 

r  v 


mo      duj.       d 
~^~dt   +  dt 


II 


+  ~dt 


mo 


1        u* 

V1-?] 


(64) 


(65) 


Introducing  the  condition  that  all  the  acceleration  is  to  be  in  the  Y 

du^- 
ction,  which  makes  -rr  =  0,  and  further  noting  t 

by  the  division  of  equation  (64)  by  (65),  we  obtain 


^- 
direction,  which  makes  -rr  =  0,  and  further  noting  that  u2  =  ux2  +  uy2, 


Fx  = 


UXU 


c2  - 


Fv. 


(66) 


Hence,  in  order  to  accelerate  a  particle  in  a  given  direction,  we  may 
apply  any  force  Fy  in  the  desired  direction,  but  must  at  the  same  time 
apply  at  right  angles  another  force  Fx  whose  magnitude  is  given  by 
equation  (66). 

Although  at  first  sight  this  state  of  affairs  might  seem  rather 
unexpected,  a  simple  qualitative  consideration  will  show  the  necessity 
of  a  component  of  force  perpendicular  to  the  desired  acceleration. 
Refer  again  to  figure  12;  since  the  particle  is  being  accelerated  in  the  Y 
direction,  its  total  velocity  and  hence  its  mass  are  increasing.  This 
increasing  mass  is  accompanied  by  increasing  momentum  in  the  X 
direction  even  when  the  velocity  in  that  direction  remains  constant. 
The  component  force  Fx  is  necessary  for  the  production  of  this  increase 
in  X-momentum. 

In  a  later  paragraph  we  shall  show  an  application  of  equation  (66) 
in  electrical  theory. 


76  Chapter  Six. 

Transverse  and  Longitudinal  Acceleration. 

69.  An  examination  of  equation  (66)  shows  that  there  are  two 
special  cases  in  which  the  component  force  Fx  disappears  and  the 
force  and  acceleration  are  in  the  same  direction.  Fx  will  disappear 
when  either  ux  or  uy  is  equal  to  zero,  so  that  force  and  acceleration 
will  be  in  the  same  direction  when  the  force  acts  exactly  at  right 
angles  to  the  line  of  motion  of  the  particle,  or  in  the  direction  of  the 
motion  (or  of  course  also  when  ux  and  uy  are  both  equal  to  zero  and 
the  particle  is  at  rest).  It  is  instructive  to  obtain  simplified  ex- 
pressions for  force  for  these  two  cases  of  transverse  and  longitudinal 
acceleration. 

Let  us  again  examine  our  equation  (60)  for  the  force  acting  on  a 
particle 

mo       du  m0  u  du 


For  the  case  of  a  transverse  acceleration  there  is  no  component  of 
force  in  the  direction  of  the  velocity  u  and  the  second  term  of  the 
equation  is  equal  to  zero,  giving  us 


For  the  case  of  longitudinal  acceleration,  the  velocity  u  and  the 

du 
acceleration  —r  are  in  the  same  direction,  so  that  we  may  rewrite  the 

second  term  of  (60),  giving  us 

„  m0       du  mo          u2  du 

T±         I 


dt  '2  fi2  dt 


and  on  simplification  this  becomes 

>$•  (68) 


Dynamics  of  a  Particle.  77 

An  examination  of  this  expression  shows  the  reason  why 


is  sometimes  spoken  of  as  the  expression  for  the  longitudinal  mass  of  a 

particle. 

The  Force  Exerted  by  a  Moving  Charge. 

70.  In  a  later  chapter  we  shah1  present  a  consistent  development 
of  the  fundamentals  of  electromagnetic  theory  based  on  the  Einstein 
transformation  equations  for  space  and  time  and  the  four  field  equa- 
tions. At  this  point,  however,  it  may  not  be  amiss  to  point  out  that 
the  principles  of  mechanics  themselves  may  sometimes  be  employed 
to  obtain  a  simple  and  direct  solution  of  electrical  problems. 

Suppose,  for  example,  we  wish  to  calculate  the  force  with  which  a 
point  charge  in  uniform  motion  acts  on  any  other  point  charge.  We 
can  solve  this  problem  by  considering  a  system  of  coordinates  which 
move  with  the  same  velocity  as  the  charge  itself.  An  observer 
making  use  of  the  new  system  of  coordinates  could  evidently  calcu- 
late the  force  exerted  by  the  charge  in  question  by  Coulomb's  familiar 
inverse  square  law  for  static  charges,  and  the  magnitude  of  the  force 
as  measured  in  the  original  system  of  coordinates  can  then  be  deter- 
mined from  our  transformation  equations  for  force.  Let  us  proceed 
to  the  specific  solution  of  the  problem. 

Consider  a  system  of  coordinates  S,  and  a  charge  e  in  uniform 
motion  along  the  X  axis  with  the  velocity  V.  We  desire  to  know 
the  force  acting  at  the  time  t  on  any  other  charge  e\  which  has  any 
desired  coordinates  x,  y,  and  z  and  any  desired  velocity  ux,  uv  and  uz. 

Assume  a  system  of  coordinates,  S',  moving  with  the  same  velocity 
as  the  charge  e  which  is  taken  coincident  with  the  origin.  To  an 
observer  moving  with  the  system  S',  the  charge  e  appears  to  be 
always  at  rest  and  surrounded  by  a  pure  electrostatic  field.  Hence 
in  system  S'  the  force  with  which  e  acts  on  e\  will  be,  in  accordance 
with  Coulomb's  law* 

eeiTr 


F' 


r'3 


*  It  should  be  noted  that  in  its  original  form  Coulomb's  law  merely  stated 
that  the  force  between  two  stationary  charges  was  proportional  to  the  product  of 
the  charges  and  inversely  to  the  distance  between  them.  In  the  present  derivation 


78  Chapter  Six. 

or 


m) 


where  xf,  y',  and  zr  are  the  coordinates  of  the  charge  e\  at  the  time  t' '. 
For  simplicity  let  us  consider  the  force  at  the  time  t'  =  0;  then  from 
transformation  equations  (9),  (10),  (11),  (12)  we  shall  have 

x'  =  KTIX,         y'  =  y,         z'  =  z. 

Substituting  in  (69),  (70),  (71)  and  also  using  our  transformation 
equations  for  force  (61),  (62),  (63),  we  obtain  the  following  equations 
for  the  force  acting  on  e^  as  it  appears  to  an  observer  in  system  S: 

^(x+5^4-^)),  (72) 


+  if  +  z 
uxV 


These  equations  give  the  force  acting  on  e\  at  the  time  t.     From 

V 
transformation  equation  (12)  we  have  t  =  —  x,  since  t'  =  0.     At  this 

time  the  charge  e,  which  is  moving  with  the  uniform  velocity  V  along 

we  have  extended  this  law  to  apply  to  the  instantaneous  force  exerted  by  a  stationary 
charge  upon  any  other  charge. 

The  fact  that  a  charge  of  electricity  appears  the  same  to  observers  in  all  systems 
is  obviously  also  necessary  for  the  setting  up  of  equations  (69),  (70),  (71).  That 
such  is  the  case,  however,  is  an  evident  consequence  of  the  atomic  nature  of  elec- 
tricity. The  charge  e  would  appear  of  the  same  magnitude  to  observers  both  in 
system  S  and  system  AS',  since  they  would  both  count  the  same  number  of  electrons 
on  the  charge.  (See  Section  157.) 


Dynamics  of  a  Particle.  79 

the  X  axis,  will  evidently  have  the  position 

V2 
xe  =—x,         ye  =  0,         ze  =  0. 

For  convenience  we  may  now  refer  our  results  to  a  system  of 
coordinates  whose  origin  coincides  with  the  position  of  the  charge  e 
at  the  instant  under  consideration.  If  X,  Y  and  Z  are  the  coordi- 
nates of  61  with  respect  to  this  new  system,  we  shall  evidently  have 
the  relations 

72 
X  =  x  -  --  x  =  K~2x,         Y  =  y,         Z  =  z, 

Ux  =  ux,         Uv  =  uv,         Uz  =  uz. 
Substituting  into  (72),  (73),  (74)  we  obtain 


(75) 

-?)(-¥)*. 


where  for  simplicity  we  have  placed 


s  = 


These  are  the  same  equations  which  would  be  obtained  by  sub- 
stituting the  well-known  formula,  for  the  strength  of  the  electric  and 
magnetic  field  around  a  moving  point  charge  into  the  fifth  funda- 
mental equation  of  the  Maxwell-Lorentz  theory,  f  =  p  (  e  +  -[u  X  h]  *  1 . 

They  are  really  obtained  in  this  way  more  easily,  however,  and  are 
seen  to  come  directly  from  Coulomb's  law. 

The  Field  around  a  Moving  Charge.  Evidently  we  may  also  use 
these  considerations  to  obtain  an  expression  for  the  electric  field 
produced  by  a  moving  charge  e,  if  we  consider  the  particular  case 
that  the  charge  ei  is  stationary  (i.  e.,  Ux  =  Uv  =  Uz  =  0)  and  equal 


80  Chapter  Six. 

to  unity.  Making  these  substitutions  in  (75),  (76),  (77)  we  obtain 
the  well-known  expression  for  the  electrical  field  in  the  neighborhood 
of  a  moving  point  charge 

where 

r  =  Xi  +  Fj  +  Zk. 

71.  Application  to  a  Specific  Problem.     Equations  (75),  (76),  (77) 

can  also  be  applied  in  the  solution  of  a 
rather  interesting  specific  problem. 

Consider  a  charge  e  constrained  to 
move  in  the  X  direction  with  the  ve- 
locity V  and  at  the  instant  under  con- 
sideration let  it  coincide  with  the  origin 
of  a  system  of  stationary  coordinates 
YeX  (fig.  13).  Suppose  now  a  second 
charge  e\,  situated  at  the  point  X  =  0, 
Y  =  Y  and  moving  in  the  X  direction 
with  the  same  velocity  V  as  the  charge  e, 
and  also  having  a  component  velocity 
in  the  F  direction  Uy.  Let  us  predict 

eQ- ,    the  nature  of  its  motion  under  the  influ- 

FlQ   13  ence  of  the  charge  e,  it  being  otherwise 

unconstrained. 

From  the  simple  qualitative  considerations  placed  at  our  disposal 
by  the  theory  of  relativity,  it  seems  evident  that  the  charge  ei  ought 
merely  to  increase  its  component  of  velocity  in  the  F  direction  and 
retain  unchanged  its  component  in  the  X  direction,  since  from  the 
point  of  view  of  an  observer  moving  along  with  e  the  phenomenon  is 
merely  one  of  ordinary  electrostatic  repulsion. 

Let  us  see  whether  our  equations  for  the  force  exerted  by  a  moving 
charge  actually  lead  to  this  result.  By  making  the  obvious  sub- 
stitutions in  equations  (75)  and  (76)  we  obtain  for  the  component 
forces  on  e\ 

„„_   /  T72\   T7 

(79) 
(80) 


Dynamics  of  a  Particle.  81 

Now  under  the  action  of  the  component  force  Fx  we  might  at 
first  sight  expect  the  charge  e\  to  obtain  an  acceleration  in  the  X 
direction,  in  contradiction  to  the  simple  qualitative  prediction  that 
we  have  just  made  on  the  basis  of  the  theory  of  relativity.  We 
remember,  however,  that  equation  (66)  prescribes  a  definite  ratio 
between  the  component  forces  Fx  and  Fv  if  the  acceleration  is  to  be 
in  the  Y  direction,  and  dividing  (79)  by  (80)  we  actually  obtain  the 
necessary  relation 

F,  _     VUV 

Fv~  c2  -  72' 

Other  applications  of  the  new  principles  of  dynamics  to  electrical, 
magnetic  and  gravitational  problems  will  be  evident  to  the  reader. 

Work. 

72.  Before  proceeding  with  the  further  development  of  our  theory 
of  dynamics  we  shall  find  it  desirable  to  define  the  quantities  work, 
kinetic,  and  potential  energy. 

We  have  already  obtained  an  expression  for  the  force  acting  on  a 
particle  and  shall  define  the  work  done  on  the  particle  as  the  integral 
of  the  force  times  the  distance  through  which  the  particle  is  dis- 
placed. Thus 

(81) 


where  r  is  the  radius  vector  determining  the  position  of  the  particle. 

Kinetic  Energy.  ' 

73.  When  a  particle  is  brought  from  a  state  of  rest  to  the  velocity 
u  by  the  action  of  an  unbalanced  force  F,  we  shall  define  its  kinetic 
energy  as  numerically  equal  to  the  work  done  in  producing  the  velocity. 
Thus 

K=  W  = 


Since,  however,  the  kinetic  energy  of  a  particle  turns  out  to  be 
entirely  independent  of  the  particular  choice  of  forces  used  in  pro- 
ducing the  final  velocity,  it  is  much  more  useful  to  have  an  expression 
for  kinetic  energy  in  terms  of  the  mass  and  velocity  of  the  particle. 

We  have 


*"/ 


F-dr  =    I    F-  -r-dt  =    I  F-udt. 


82 


Chapter  Six. 


Substituting  the  value  of  F  given  by  the  equation  of  definition  (47) 
we  obtain 


K 


/du  Cdm 

m-j-  -udt  +       -rru 
dt  J    dt 

=    I  mu  •  du  +   I  u  •  udm 
=    I  mudu  +   I  uzdm. 


udt 


Introducing  the  expression  (51)  for  the  mass  of  a  moving  particle 

W0 


m  = 


= ,  we  obtain 


and  on  integrating  and  evaluating  the  constant  of  integration  by 
placing  the  kinetic  energy  equal  to  zero  when  the  velocity  is  zero, 
we  easily  obtain  the  desired  expression  for  the  kinetic  energy  of  a 
particle : 

1 


K  =  m0c2 


=  c2(m  —  WQ). 


(82) 


(83) 


It  should  be  noticed,  as  was  stated  above,  that  the  kinetic  energy 
of  a  particle  does  depend  merely  on  its  mass  and  final  velocity  and  is 
entirely  independent  of  the  particular  choice  of  forces  which  happened 
to  be  used  in  producing  the  state  of  motion. 

It  will  also  be  noticed,  on  expansion  into  a  series,  that  our  ex- 
pression (82)  for  the  kinetic  energy  of  a  particle  approaches  at  low 
velocities  the  form  familiar  in  the  older  Newtonian  mechanics, 

K  —  Jwo^2. 
Potential  Energy. 

74.  When  a  moving  particle  is  brought  to  rest  by  the  action  of  a 


Dynamics  of  a  Particle.  83 

conservative*  force  we  say  that  its  kinetic  energy  has  been  trans- 
formed into  potential  energy.  The  increase  in  the  potential  energy 
of  the  particle  is  equal  to  the  kinetic  energy  which  has  been  destroyed 
and  hence  equal  to  the  work  done  by  the  particle  against  the  force, 
giving  us  the  equation 

--  W  =  --F-dr.  (84) 


The  Relation  between  Mass  and  Energy. 

75.  We  may  now  consider  a  very  important  relation  between  the 
mass  and  energy  of  a  particle,  which  was  first  pointed  out  in  our 
chapter  on  "  Some  Elementary  Deductions." 

When  an  isolated  particle  is  set  in  motion,  both  its  mass  and 
energy  are  increased.  For  the  increase  in  mass  we  may  write 

Am  =  ra  —  m0, 

and  for  the  increase  in  energy  we  have  the  expression  for  kinetic  energy 
given  in  equation  (83),  giving  us 

AE  —  c2(m  —  mo), 
or,  combining  with  the  previous  equation, 

AE  =  <Mm.  (85) 

Thus  the  increase  in  the  kinetic  energy  of  a  particle  always  bears 
the  same  definite  ratio  (the  square  of  the  velocity  of  light)  to  its 
increase  in  mass.  Furthermore,  when  a  moving  particle  is  brought 
to  rest  and  thus  loses  both  its  kinetic  energy  and  its  extra  ("  kinetic  ") 
mass,  there  seems  to  be  every  reason  for  believing  that  this  mass 
and  energy  which  are  associated  together  when  the  particle  is  in 
motion  and  leave  the  particle  when  it  is  brought  to  rest  will  still 
remain  alwaj^s  associated  together.  For  example,  if  the  particle  is 
brought  to  rest  by  collision  with  another  particle,  it  is  an  evident 

*  A  conservative  force  is  one  such  that  any  work  done  by  displacing  a  system 
against  it  would  be  completely  regained  if  the  motion  of  the  system  should  be  re- 
versed. 

Since  we  believe  that  the  forces  which  act  on  the  ultimate  particles  and  con- 
stituents of  matter  are  in  reality  all  of  them  conservative,  we  shall  accept  the  general 
principle  of  the  conservation  of  energy  just  as  in  Newtonian  mechanics.  (For  a 
logical  deduction  of  the  principle  of  the  conservation  of  energy  in  a  system  of  par- 
ticles, see  the  next  chapter,  section  89.) 


84  Chapter  Six. 

consequence  of  our  considerations  that  the  energy  and  the  mass 
corresponding  to  it  do  remain  associated  together  since  they  are  both 
passed  on  to  the  new  particle.  On  the  other  hand,  if  the  particle 
is  brought  to  rest  by  the  action  of  a  conservative  force,  say  for  example 
that  exerted  by  an  elastic  spring,  the  kinetic  energy  which  leaves  the 
particle  will  be  transformed  into  the  potential  energy  of  the  stretched 
spring,  and  since  the  mass  which  has  undoubtedly  left  the  particle 
must  still  be  in  existence,  we  shall  believe  that  this  mass  is  now  asso- 
ciated with  the  potential  energy  of  the  stretched  spring. 

76.  Such  considerations  have  led  us  to  believe  that  matter  and 
energy  may  be  best  regarded  as  different  names  for  the  same  funda- 
mental entity:  matter,  the  name  which  has  been  applied  when  we 
have  been  interested  in  the  property  of  mass  or  inertia  possessed 
by  the  entity,  and  energy,  the  name  applied  when  we  have  been 
interested  in  the  part  taken  by  the  entity  in  the  production  of  motion 
and  other  changes  in  the  physical  universe.     We  shall  find  these 
ideas  as  to  the  relations  between  matter,  energy  and  mass  very  fruit- 
ful in  the  simplification  of  physical  reasoning,  not  only  because  it 
identifies  the  two  laws  of  the  conservation  of  mass  and  the  conser- 
vation of  energy,  but  also  for  its  frequent  application  in  the  solution 
of  specific  problems. 

77.  We  must  call  attention  to  the  great  difference  in  size  between 
the  two  units,  the  gram  and  the  erg,  both  of  which  are  used  for  the 
measurement  of  the  one  fundamental  entity,  call  it  matter  or  energy 
as  we  please.     Equation  (85)  gives  us  the  relation 

E  =  c*m,  (86) 

where  E  is  expressed  in  ergs  and  m  in  grams;  hence,  taking  the  velocity 
of  light  as  3  X  1010  centimeters  per  second,  we  shall  have 

1  gram  =  9  X  1020  ergs.  (87) 

The  enormous  number  of  ergs  necessary  for  increasing  the  mass  of 
a  system  to  the  amount  of  a  single  gram  makes  it  evident  that  experi- 
mental proofs  of  the  relation  between  mass  and  energy  will  be  hard  to 
find,  and  outside  of  the  experimental  work  on  electrons  at  high  veloci- 
ties, already  mentioned  in  Section  64  and  the  well-known  relations 


Dynamics  of  a  Particle.  85 

between  the  energy  and  momentum  of  a  beam  of  light,  such  evidence 
has  not  yet  been  forthcoming. 

As  to  the  possibility  of  obtaining  further  direct  experimental 
evidence  of  the  relation  between  mass  and  energy,  we  certainly  can- 
not look  towards  thermal  experiments  with  any  degree  of  confidence, 
since  even  on  cooling  a  body  down  to  the  absolute  zero  of  temperature 
it  loses  but  an  inappreciable  fraction  of  its  mass  at  ordinary  tempera- 
tures.* In  the  case  of  some  radioactive  processes,  however,  we  may 
find  a  transfer  of  energy  large  enough  to  bring  about  measurable 
differences  in  mass.  And  making  use  of  this  point  of  view  we  might 
account  for  the  lack  of  exact  relations  between  the  atomic  weights  of 
the  successive  products  of  radioactive  decomposition,  f 

78.  Application  to  a  Specific  Problem.  We  may  show  an  inter- 
esting application  of  our  ideas  as  to  the  relation  between  mass  and 
energy,  in  the  treatment  of  a  specific  problem.  Consider,  just  as  in 
Section  63,  two  elastic  particles  both  of  which  have  the  mass  ra0  at 
rest,  one  moving  in  the  X  direction  with  the  velocity  +  u  and  the 
other  with  the  velocity  —  w,  in  such  a  way  that  a  head-on  collision 
between  the  particles  will  occur  and  they  will  rebound  over  their 
original  paths  with  the  respective  velocities  —  u  and  +  u  of  the 
same  magnitude  as  before. 

Let  us  now  consider  how  this  collision  would  appear  to  an  observer 
who  is  moving  past  the  original  system  of  coordinates  with  the  velocity 
V  in  the  X  direction.  To  this  new  observer  the  particles  will  be 
moving  before  the  collision  with  the  respective  velocities 

u  —  V  -  u  -  V 

Ul  =       —          and         u2=         -—, 


as  given  by  equation  (14)  for  the  transformation  of  velocities.  Fur- 
thermore. when  in  the  course  of  the  collision  the  particles  have  come 
to  relative  rest  they  will  obviously  be  moving  past  our  observer  with 
the  velocity  —  V. 

*  It  should  be  noticed  that  our  theory  points  to  the  presence  of  enormous. 
stores  of  interatomic  energy  which  are  still  left  in  substances  cooled  to  the  absolute 
zero. 

t  See,  for  example,  Comstock,  Philosophical  Magazine,  vol.  15,  p.  1  (1908). 


86 


Chapter  Six. 


Let  us  see  what  the  masses  of  the  particles  will  be  both  before  and 
during  the  collision.  Before  the  collision,  the  mass  of  the  first  particle 
will  be 


w0 


W0 


u-V 

_uV 
1  ~   c2 


and  the  mass  of  the  second  particle  will  be 


W0 


W0 


W0 


Adding  these  two  expressions,  we  obtain  for  the  sum  of  the  masses  of 
the  two  particles  before  collision, 

2w0 


Now  during  the  collision,  when  the  two  particles  have  come  to 
relative  rest,  they  will  evidently  both  be  moving  past  our  observer 
with  the  velocity  —  V  and  hence  the  sum  of  their  masses  at  the 
instant  of  relative  rest  would  appear  to  be 

2w0 


a  quantity  which  is  smaller  than  that  which  we  have  just  found  for 
the  sum  of  the  two  masses  before  the  collision  occurred.  This  apparent 
discrepancy  between  the  total  mass  of  the  system  before  and  during 
the  collision,  is  removed,  however,  if  we  realize  that  when  the  par- 


Dynamics  of  a  Particle.  87 

tides  have  come  to  relative  rest  an  amount  of  potential  energy  of 
elastic  deformation  has  been  produced,  which  is  just  sufficient  to  re- 
store them  to  their  original  velocities,  and  the  mass  corresponding  to 
this  potential  energy  will  evidently  be  just  sufficient  to  make  the 
total  mass  of  the  system  the  same  as  before  collision. 

In  the  following  chapter  on  the  dynamics  of  a  system  of  particles 
we  shall  make  further  use  of  our  ideas  as  to  the  mass  corresponding 
to  potential  energy. 


CHAPTER  VII. 


THE  DYNAMICS  OF  A  SYSTEM   OF  PARTICLES. 

79.  In  the  preceding  chapter  we  discussed  the  laws  of  motion 
of  a  particle.     With  the  help  of  those  laws  we  shall  now  derive  some 
useful  general  dynamical  principles  which  describe  the  motions  of  a 
system  of  particles,  and  in  the  following  chapter  shall  consider  an 
application  of  some  of  these  principles  to  the  kinetic  theory  of  gases. 

The  general  dynamical  principles  which  we  shall  present  in  this 
chapter  will  be  similar  in  form  to  principles  which  are  already  familiar 
in  the  classical  Newtonian  mechanics.  Thus  we  shall  deduce  princi- 
ples corresponding  to  the  principles  of  the  conservation  of  momentum, 
of  the  conservation  of  moment  of  momentum,  of  least  action  and  of 
vis  viva,  as  well  as  the  equations  of  motion  in  the  Lagrangian  and 
Hamiltonian  (canonical)  forms.  For  cases  where  the  velocities  of  all 
the  particles  involved  are  slow  compared  with  that  of  light,  we  shall 
find,  moreover,  that  our  principles  become  identical  in  content,  as 
well  as  in  form,  with  the  corresponding  principles  of  the  classical 
mechanics.  Where  high  velocities  are  involved,  however,  our  new 
principles  will  differ  from  those  of  Newtonian  mechanics.  In  par- 
ticular we  shall  find  among  other  differences  that  in  the  case  of  high 
velocities  it  will  no  longer  be  possible  to  define  the  Lagrangian  function 
as  the  difference  between  the  kinetic  and  potential  energies  of  the 
system,  nor  to  define  the  generalized  momenta  used  in  the  Hamil- 
tonian equations  as  the  partial  differential  of  the  kinetic  energy  with 
respect  to  the  generalized  velocity. 

On  the  Nature  of  a  System  of  Particles. 

80.  Our  purpose  in  this  chapter  is  to  treat  dynamical  systems 
consisting  of  a  finite  number  of  particles,  each  obeying  the  equation 
of  motion  which  we  have  already  written  in  the  forms, 


-   d 


du 


dm 
~dt 


F  = 


dt 


WQ 


WQ       du       d 
^~dt  +  dt 


88 


u. 


(47) 
(59) 


Dynamics  of  a  System  of  Particles.  89 

It  is  not  to  be  supposed,  however,  that  the  total  mass  of  such  a 
system  can  be  taken  as  located  solely  in  these  particles.  It  is  evident 
rather,  since  potential  energy  has  mass,  that  there  will  in  general  be 
mass  distributed  more  or  less  continuously  throughout  the  space  in 
the  neighborhood  of  the  particles.  Indeed  we  have  shown  at  the 
end  of  the  preceding  chapter  (Section  78)  that  unless  we  take  account 
of  the  mass  corresponding  to  potential  energy  we  can  not  maintain 
the  principle  of  the  conservation  of  mass,  and  we  should  also  find  it 
impossible  to  retain  the  principle  of  the  conservation  of  momentum 
unless  we  included  the  momentum  corresponding  to  potential  energy. 

For  a  continuous  distribution  of  mass  we  may  write  for  the  force 
acting  at  any  point  on  the  material  in  a  small  volume,  57, 

fS7=|(gS7),  (47  A) 

where  f  is  the  force  per  unit  volume  and  g  is  the  density  of  momentum. 
This  equation  is  of  course  merely  an  equation  of  definition  for  the 
intensity  of  force  at  a  point.  We  shall  assume,  however,  that  New- 
ton's third  law,  that  is,  the  principle  of  the  equality  of  action  and 
reaction,  holds  for  forces  of  this  type  as  well  as  for  those  acting  on 
particles.  In  later  chapters  we  shall  investigate  the  way  in  which  g 
depends  on  velocity,  state  of  strain,  etc.,  but  for  the  purposes  of  this 
chapter  we  shall  not  need  any  further  information  as  to  the  nature 
of  the  distributed  momentum. 

Let  us  proceed  to  the  solution  of  our  specific  problems. 

The  Conservation  of  Momentum. 

81.  We  may  first  show  from  Newton's  third  law  of  motion  that 
the  momentum  of  an  isolated  system  of  particles  remains  constant. 

Considering  a  system  of  particles  of  masses  Wi,  ra2,  ra3,  etc.,  we 
may  write  in  accordance  with  equation  47, 

_  d 

d  (89) 

F2  +  I2  =  jt  ' 

etc., 


90  Chapter  Seven. 

where  FI,  F2,  etc.,  are  the  external  forces  impressed  on  the  individual 
particles  from  outside  the  system  and  Ii,  I2;  etc.,  are  the  internal 
forces  arising  from  mutual  reactions  within  the  interior  of  the  system. 
Considering  the  distributed  mass  in  the  system,  we  may  also  write, 
in  accordance  with  (47 A)  the  further  equation 

(f  4-1)57  =^(g57),  (90) 

where  f  and  i  are  respectively  the  external  and  internal  forces  acting 
per  unit  volume  of  the  distributed  mass.  Integrating  throughout  the 
whole  volume  of  the  system  V  we  have 

=  Tt>  (91) 

where  G  is  the  total  distributed  momentum  in  the  system.  Adding 
this  to  our  previous  equations  (89)  for  the  forces  acting  on  the  indi- 
vidual particles,  we  have 


+  Zli  +  j*fdV  +  fidV  =  ^ 


+ 


But  from  Newton's  third  law  of  motion  (i.  e.,  the  principle  of  the 
equality  of  action  and  reaction)  it  is  evident  that  the  sum  of  the 

internal  forces,  Zli  +  J  idV,  which  arise  from  mutual  reactions  within 
the  system  must  be  equal  to  zero,  which  leads  to  the  desired  equation 
of  momentum 

C  d 

ZFi  +    I  tdo  =  Jt  (SwiiUi  -f  G).  (92) 

In  words  this  equation  states  that  at  any  given  instant  the  vector 
sum  of  the  external  forces  acting  on  the  system  is  equal  to  the  rate 
at  which  the  total  momentum  of  the  system  is  changing. 

For  the  particular  case  of  an  isolated  system  there  are  no  external 
forces  and  our  equation  becomes  a  statement  of  the  principle  of  the 
conservation  of  momentum. 
The  Equation  of  Angular  Momentum. 

82.  We  may  next  obtain  an  equation  for  the  moment  of  momentum 
of  a  system  about  a  point. 


Dynamics  of  a  System  of  Particles.  91 

Consider  a  particle  of  mass  Wi  and  velocity  Ui.  Let  TI  be  the 
radius  vector  from  any  given  point  of  reference  to  the  particle.  Then 
for  the  moment  of  momentum  of  the  particle  about  the  point  we  may 
write 

MI  =  TI  X 


and  summing  up  for  all  the  particles  of  the  system  we  may  write 

=  2(ri  X  miuO.  (93) 


Similarly,  for  the  moment  of  momentum  of  the  distributed  mass  we 
may  write 

Mdist.  =  /  (r  X  g)dV,  (94) 

where  r  is  the  radius  vector  from  our  chosen  point  of  reference  to  a 
point  in  space  where  the  density  of  momentum  is  g  and  the  inte- 
gration is  to  be  taken  throughout  the  whole  volume,  V,  of  the  system. 
Adding  these  two  equations  (93)  and  (94),  we  obtain  for  the  total 
amount  of  momentum  of  the  system  about  our  chosen  point 


M  =  2(f!  X  TKiUi)  +  /  (r  X  g)dV', 


and  differentiating  with  respect  to  the  time  we  have,  for  the  rate  of 
change  of  the  moment  of  momentum, 


or,  making  the  substitutions  given  by  equations  (89)  and  (90),  and 
writing  -~  =  Ui,  etc.  we  have 

~  =  2(rx  X  FO  +  2(rx  X  Ii)  +  2(ui  X  wiiii) 

+  /  (r  X  f)dV  +  /  (r  X  i)dV  +  f  (u  X  g)dV. 

To  simplify  this  equation  we  may  note  that  the  third  term  is  equal  to 
zero  because  it  contains  the  outer  product  of  a  vector  by  itself.  Fur- 
thermore, if  we  accept  the  principle  of  the  equality  of  action  and 


92  Chapter  Seven. 

reaction,  together  with  the  further  requirement  that  forces  are  not 
only  equal  and  opposite  but  that  their  points  of  application  be  in  the 
same  straight  line,  we  may  put  the  moment  of  all  the  internal  forces 
equal  to  zero  and  thus  eliminate  the  second  and  fifth  terms.  We 
obtain  as  the  equation  of  angular  momentum 


X  FO  +  J  (r  X  f)dV  +  JVx  g)dV.         (95) 


*  We  may  call  attention  to  the  inclusion  in  this  equation  of  the 
interesting  term  J  (u  X  g)dV.  If  density  of  momentum  and  velocity 
should  always  be  in  the  same  direction  this  term  would  vanish,  since 
the  outer  product  of  a  vector  by  itself  is  equal  to  zero.  In  our  con- 
sideration of  the  "  Dynamics  of  Elastic  Bodies,"  however,  we  shall 
find  bodies  with  a  component  of  momentum  at  right  angles  to  their 
direction  of  motion  and  hence  must  include  this  term  in  a  general 
treatment.  For  a  completely  isolated  system  it  can  be  shown,  how- 
ever, that  this  term  vanishes  along  with  the  external  forces  and  we 
then  have  the  principle  of  the  conservation  of  moment  of  momentum. 
The  Function  T. 

83.  We  may  now  proceed  to  the  definition  of  a  function  which 
will  be  needed  in  our  treatment  of  the  principle  of  least  action. 

One  of  the  most  valuable  properties  of  the  Newtonian  expression, 
^m0w2,  for  kinetic  energy  was  the  fact  that  its  derivative  with  respect 
to  velocity  is  evidently  the  Newtonian  expression  for  momentum,  mQu* 
It  is  not  true,  however,  that  the  derivative  of  our  new  expression 

1 


for  kinetic  energy  (see  Section  73),  m0c5 


-  1 


,  with  respect 


to  velocity  is  equal  to  momentum,  and  for  that  reason  in  our  non- 
Newtonian  mechanics  we  shall  find  it  desirable  to  define  a  new  func- 
tion, Ty  by  the  equation, 

/  /  /i,2\ 

(96) 

For  slow  velocities  (i.  e.,  small  values  of  u)  this  reduces  to  the 
Newtonian  expression  for  kinetic  energy  and  at  all  velocities  we  have 


Dynamics  of  a  System  of  Particles.  93 

the  relation, 


d     I        v? 
Tu  V1  -  ?  = 


dT  .  d     /        u?          m0u 


showing  that  the  differential  of  T  with  respect  to  velocity  is  momentum. 
For  a  system  of  particles  we  shall  define  T  as  the  summation  of 
the  values  for  the  individual  particles: 


(98) 


The  Modified  Lagrangian  Function. 

84.  In  the  older  mechanics  the  Lagrangian  function  for  a  system 
of  particles  was  defined  as  the  difference  between  the  kinetic  and 
potential  energies  of  the  system.     The  value  of  the  definition  rested, 
however,  on  the  fact  that  the  differential  of  the  kinetic  energy  with 
respect  to  velocity  was  equal  to  momentum,  so  that  we  shall  now 
find  it  advisable  to  define  the  Lagrangian  function  with  the  help  of 
our  new  function  T  in  accordance  with  the  equation 

L  =  T  -  U.  (99) 

The  Principle  of  Least  Action. 

85.  We  are  now  in  a  position  to.  derive  a  principle  corresponding 
to  that  of  least  action  in  the  older  mechanics.     Consider  the  path 
by  which  our  dynamical  system  actually  moves  from  state  (1)  to 
state  (2).     The  motion  of  any  particle  in  the  system  of  mass  m  will 
be  governed  by  the  equation 

P  - 1  (mu).  (100) 

Let  us  now  compare  the  actual  path  by  which  the  system  moves 
from  state  (1)  to  state  (2)  with  a  slightly  displaced  path  in  which  the 
laws  of  motion  are  not  obeyed,  and  let  the  displacement  of  the  particle 
at  the  instant  in  question  be  dr. 

Let  us  take  the  inner  product  of  both  sides  of  equation  (100)  with 


94  Chapter  Seven. 

5r;  we  have 


d_ 
It 

d  ,  ddr 


F-5r  =  -j-  (wu)-Sr 


rf 

~  dt 

(rau-Su  +  ¥-8r)dt  =  d(wu-5r). 

Summing  up  for  all  the  particles  of  the  system  and  integrating 
between  the  limits  ti  and  t%,  we  have 


Since  £1  and  tz  are  the  times  when  the  actual  and  displaced  motions 
coincide,  we  have  at  these  times  5r  =  0;  furthermore  we  also  have 
u  •  Su  =  udu,  so  that  we  may  write 


f 


(Zmubu  +  F-Sr)d*  =  0. 


With  the  help  of  equation  (97),  however,  we  see  that  Zmudu  =  dT, 
giving  us 

•<2 

(dT  +.¥-8r)dt  =  0.  (101) 

//  the  forces  F  are  conservative,  we  may  write  F-5r  =  —  8U,  where 
dU  is  the  difference  between  the  potential  energies  of  the  displaced 
and  the  actual  configurations.  This  gives  us 


U)dt  =  0 
•/«, 

or 


/»«! 

(T- 
Jt, 


f 
J« 


JxB  «  0,  (102) 


which  is  the  modified  principle  of  least  action.     The  principle  evi- 
dently requires  that  for  the  actual  path  by  which  the  system  goes 


Dynamics  of  a  System  of  Particles.  95 

/•«• 

from  state  (1)  to  state  (2),  the  quantity  I     Ldt  shall  be  a  minimum  (or 

Jtl 

maximum). 

Lagrange's  Equations. 

86.  We  may  now  derive  the  Lagrangian  equations  of  motion  from 
the  above  principle  of  least  action.  Let  us  suppose  that  the  position 
of  each  particle  of  the  system  under  consideration  is  completely  deter- 
mined by  n  independent  generalized  coordinates  0i,  02,  0s  •  •  •  <f>n  and 
hence  that  L  is  some  function  of  0i,  02,  0s  •  •  •  0n,  <t>i,  02,  fa  •  -  •  <i>n, 


where  for  simplicity  we  have  put  0i  —  —^r  ,  02  =  ~~JT  »  etc. 
From  equation  (102)  we  have 

/»(j  x»*2    /     n       -JT  n       nT  \ 

(5L)dt=      (Zlr^  +  ^r^O*"'*0-    (103) 

Jt  Jt     \   i    d<f)i  !    d(f>i        J 


But 


which  gives  us 

dL  Ctn-  dL   d 

'iK1*-     «* 


or,  since  at  times  £1  and  £2,  50  1  is  zero,  the  first  term  in  this  expression 
disappears  and  on  substituting  in  equation  (103)  we  obtain 


Since,  however,  the  limits  ti  and  t2  are  entirely  at  our  disposal  we  must 
have  at  every  instant 

)L       d  f  di 


Finally,  moreover,  since  the  0's  are  independent  parameters,  we  can 
assign  perfectly  arbitrary  values  to  50i,  502,  etc.,  and  hence  must  have 


96  Chapter  Seven. 

the  series  of  equations 

—  (  I  — =  0 

dt\d<i>J     091 

»L  (104) 


etc. 

These  correspond  to  Lagrange's  equations  in  the  older  mechanics, 
differing  only  in  the  definition  of  L. 

Equations  of  Motion  in  the  Hamiltonian  Form. 

87.  We  shall  also  find  it  desirable  to  obtain  equations  of  motion 
in  the  Hamiltonian  or  canonical  form. 

Let  us  define  the  generalized  momentum  ^i  corresponding  to  the 
coordinate  91  by  the  equation, 

*x-.  (105) 


It  should  be  noted  that  the  generalized  momentum  is  not  as  in 
ordinary  mechanics  the  derivative  of  the  kinetic  energy  with  respect 
to  the  generalized  velocity  but  approaches  that  value  at  low  velocities. 

Consider  now  a  function  T'  defined  by  the  equation 

Tr  -  Vi9i  +  1^292  +  -  -  •  -T.  (106) 

Differentiating  we  have 

dT'  =  ii 


dT  dT 

—  TT~  »w  ~"  ->  r  ^^2  —  •  • 
091  092 

dT  ,       .dT 

—  vr  »9i  —  vr  «^2  —  •  •  • 

u<pi  C/92 

and  this,  by  the  introduction  of  (105),  becomes 

/•IT7  *\  fTI 

dT'  =  91^1  +  92^2  +  •  •  •  -  vr^9i  -     r^92  -••..-.      (107) 

091 


Dynamics  of  a  System  of  Particles.  97 

Examining  this  equation  we  have 

dT'  dT 

$</>i  d<f>i 

fttpt 

^  =  *i-  (109) 

In  Lagrange's  equations  we  have 


But  since  U  is  independent  of  \f/i  we  may  write 

d(T  -  U)  _    dT 
d<i>i 

and  furthermore  by  (108), 


Substituting  these  two  expressions  in  Lagrange's  equations  we  obtain 

1  d(T'  +  U) 


dt 
or,  writing  T'  -f-  U  =  E,  we  have 

dE 


and  since  U  is  independent  of  \f/i  we  may  rewrite  equation  (109)  in 
the  form 

dE 


The  set  of  equations  corresponding  to  (110)  and  (111)  for  all  the 
coordinates  <£i,  <£2.  03,  •  •  •  <f>n  and  the  momenta  ^i,  ^2,  1^3,  •  •  •  ^n  are 
the  desired  equations  of  motion  in  the  canonical  form. 

88.  Value  of  the  Function  T'.  We  have  given  the  symbol  E  to 
the  quantity  T'  +  U,  since  T'  actually  turns  out  to  be  identical  with 

8 


98  Chapter  Seven. 

- 

the  expression  by  which  we  defined  kinetic  energy,  thus  making 
E  =  T"  -f-  U  the  sum  of  the  kinetic  and  potential  energies  of  the 
system. 

To  show  that  T'  is  equal  to  K,  the  kinetic  energy,  we  have  by  the 
equation  of  definition  (106) 

T'  =  Mi  +  M*  +  •  •  •  -  T, 
.  dT        .  dT 


But  T  by  definition,  equation  (98),  is 
T  = 


which  gives  us 

\         c2  /       U  d0! 


and  substituting  we  obtain 


T'  =  ^mu--  +  <t>2^mu~  +  -  -  -  -  T 


We  can  show,  however,  that  the  term  in  parenthesis  is  equal  to  u. 
If  the  coordinates  x,  y,  z  determine  the  position  of  the  particle  in 
question,  we  have, 


and  differentiating  with  respect  to  the  <£'s,  we  obtain, 

d£  =  df(  )        dx         dx_  _  dx_        dx        dx 

' 


Dynamics  ofv  System  of  Particles.  99 

Similarly 

dy        dy         dy        dy 

•i  ;     =  TT"  »       \  :     =  TTT"  '        6tC., 


jd£        62         62         dz 


Let  us  write  now 


u  =  Vx2  +  i/2  4-  22, 

dw  1 


dx      dy 

or  making  the  substitutions  for  77-  ,  77-  ,  etc.,  given  above,  we  have, 

d0i     091 


^!L  =  \(  -*L  j_  -*!L  j   •  dz  \ 

d+i"*  U\*  d+i      *  d+i      Zd<t>J' 
Substituting  now  in  (112)  we  shall  obtain, 


-  T 
or,  introducing  the  value  of  T  given  by  equation  (98),  we  have 


=  2c*(w  —  m0), 

which  is  the  expression  (83)  for  kinetic  energy. 

Hence  we  see  that  the  Hamiltonian  function  E  =  T'  -f  U  is  the 
sum  of  the  kinetic  and  potential  energies  of  the  system  as  in  Newtonian 
mechanics. 

The  Principle  of  the  Conservation  of  Energy. 

89.  We  may  now  make  use  of  our  equations  of  motion  in  the 
canonical  form  to  show  that  the  total  energy  of  a  system  of  interacting 


100  Chapter  Seven. 

particles  remains  constant.  If  such  were  not  the  case  it  is  obvious 
that  our  definitions  of  potential  and  kinetic  energy  would  not  be 
very  useful. 

Since  E  =  T'  +  U  is  a  function  of  <£i,  <£2,  <£s,  •  •  •  fa,  ^2,  i/%  •  •  •  ,  we 
may  write 

dE      dE  .        dE  . 


dE  dE 


dE      dE 

Substituting  the  values  of  -  —  .  -  —  .  etc.,  given  by  the  canonical 

o0i     d\f/i 

equations  of  motion  (110)  and  (111),  we  have 
dE 

—    =     —    \f/l<j>l    —    ^202    - 

+   $l&   +   ^2^2   +    '  '  ' 

=  0, 

which  gives  us  the  desired  proof  that  just  as  in  the  older  Newtonian 
mechanics  the  total  energy  of  an  isolated  system  of  particles  is  a 
conservative  quantity. 

On  the  Location  of  Energy  in  Space. 

90.  This  proof  of  the  conservation  of  energy  in  a  system  of  inter- 
acting particles  justifies  us  in  the  belief  that  the  concept  of  energy 
will  not  fail  to  retain  in  the  newer  mechanics  the  position  of  great 
importance  which  it  gradually  acquired  in  the  older  systems  of  physical 
theory.  Indeed,  our  newer  considerations  have  augmented  the 
important  role  of  energy  by  adding  to  its  properties  the  attribute  of 
mass  or  inertia,  and  thus  leading  to  the  further  belief  that  matter 
and  energy  are  in  reality  different  names  for  the  same  fundamental 
entity. 

The  importance  of  this  entity,  energy,  makes  it  very  interesting 
to  consider  the  possibility  of  ascribing  a  definite  location  in  space  to 
any  given  quantity  of  energy.  In  the  older  mechanics  we  had  a 
hazy  notion  that  the  kinetic  energy  of  a  moving  body  was  probably 
located  in  some  way  in  the  moving  body  itself,  and  possibly  a  vague 


Dynamics  of  a  System  of  Particles.  101 

idea  that  the  potential  energy  of  a  raised  weight  might  be  located  in 
the  space  between  the  weight  and  the  earth.  Our  discovery  of  the 
relation  between  mass  and  energy  has  made  it  possible,  however,  to 
give  a  much  more  definite,  although  not  a  complete,  answer  to  inquiries 
of  this  kind. 

In  our  discussions  of  the  dynamics  of  a  particle  (Chapter  VI, 
Section  61)  we  saw  that  an  acceptance  of  Newton's  principle  of  the 
equality  of  action  and  reaction  forced  us  to  ascribe  an  increased  mass 
to  a  moving  particle  over  that  which  it  has  at  rest.  This  increase  in 
the  mass  of  the  moving  particle  is  necessarily  located  either  in  the 
particle  itself  or  distributed  in  the  surrounding  space  in  such  a  way 
that  its  center  of  mass  always  coincides  with  the  position  of  the 
particle,  and  since  the  kinetic  energy  of  the  particle  is  the  energy 
corresponding  to  this  increased  mass  we  may  say  that  the  kinetic  energy 
of  a  moving  particle  is  so  distributed  in  space  that  its  center  of  mass 
always  coincides  with  the  position  of  the  particle. 

If  now  we  consider  the  transformation  of  kinetic  energy  into 
potential  energy  we  can  also  draw  somewhat  definite  conclusions  as  to 
the  location  of  potential  energy.  By  the  principle  of  the  conserva- 
tion of  mass  we  shall  be  able  to  say  that  the  mass  of  any  potential 
energy  formed  is  just  equal  to  the  "  kinetic  "  mass  which  has  dis- 
appeared, and  by  the  principle  of  the  conservation  of  momentum  we 
can  say  that  the  velocity  of  this  potential  energy  is  just  that  necessary 
to  keep  the  total  momentum  of  the  system  constant.  Such  con- 
siderations will  often  permit  us  to  reach  a  good  idea  as  to  the  location 
of  potential  energy. 

Consider,  for  example,  a  pair  of  similar  attracting  particles  which 
are  moving  apart  from  each  other  with  the  velocities  +  u  and  -  u 
and  are  gradually  coming  to  rest  under  the  action  of  their  mutual 
attraction,  their  kinetic  energy  thus  being  gradually  changed  into 
potential  energy.  Since  the  total  momentum  of  the  system  must 
always  remain  zero,  we  may  think  of  the  potential  energy  which  is 
formed  as  left  stationary  in  the  space  between  the  two  particles. 


CHAPTER  VIII. 
THE  CHAOTIC  MOTION  OF  A  SYSTEM   OF  PARTICLES. 

The  discussions  of  the  previous  chapter  have  placed  at  our  disposal 
generalized  equations  of  motion  for  a  system  of  particles  similar  in 
form  to  those  familiar  in  the  classical  mechanics,  and  differing  only 
in  the  definition  of  the  Lagrangian  function.  With  the  help  of  these 
equations  it  is  possible  to  carry  out  investigations  parallel  to  those 
already  developed  in  the  classical  mechanics,  and  in  the  present 
chapter  we  shall  discuss  the  chaotic  motion  of  a  system  of  particles. 
This  problem  has  received  much  attention  in  the  classical  mechanics 
because  of  the  close  relations  between  the  theoretical  behavior  of 
such  an  ideal  system  of  particles  and  the  actual  behavior  of  a  mona- 
tomic  gas.  We  shall  find  no  more  difficulty  in  handling  the  problem 
than  was  experienced  in  the  older  mechanics,  and  our  results  will  of 
course  reduce  to  those  of  Newtonian  mechanics  in  the  case  of  slow 
velocities.  Thus  we  shall  find  a  distribution  law  for  momenta  which 
reduces  to  that  of  Maxwell  for  slow  velocities,  and  an  equipartition 
law  for  the  average  value  of  a  function  which  at  low  velocities  becomes 
identical  with  the  kinetic  energy  of  the  particles. 

91.  The  Equations  of  Motion.  It  has  been  shown  that  the  Hamil- 
tonian  equations  of  motion 


dE_ 

d</>i  "     "   dt 


3E_  _ 

~  dt 


etc., 

will  hold  in  relativity  mechanics  provided  we  define  the  generalized 
momenta  \f/i,  ^2,  etc.,  not  as  the  differential  of  the  kinetic  energy 
with  respect  to  the  generalized  velocities  <£i,  <£2,  etc.,  but  as  the  dif- 
ferential with  respect  to  <£i,  <j>2)  etc.,  of  a  function 


102 


Chaotic  Motion  of  a  System  of  Particles.  103 

where  w0  is  the  mass  of  a  particle  having  the  velocity  u  and  the  sum- 
mation 2  extends  over  all  the  particles  of  the  system. 

92.  Representation  in  Generalized  Space.     Consider  now  a  system 
defined  by  the  n  generalized  coordinates  fa,  fa,  fa,  —  ',  <t>n,  and  the 
corresponding  momenta  \f/i}  \f/2,  ^3,  •  •  • ,  ^n-     Employing  the  methods 
so  successfully  used  by  Jeans,*  we  may  think  of  the  state  of  the 
system  at  any  instant  as  determined  by  the  position  of  a  point  plotted 
in  a  2n-dimensional  space.     Suppose  now  we  had  a  large  number  of 
systems  of  the  same  structure  but  differing  in  state,  then  for  each 
system  we  should  have  at  any  instant  a  corresponding  point  in  our 
2n-dimensional  space,  and  as  the  systems  changed  their  state,  in  the 
manner  required  by  the  laws  of  motion,  the  points  would  describe 
stream  lines  in  this  space. 

93.  Liouville's  Theorem.     Suppose    now    that    the    points    were 
originally   distributed   in   the   generalized   space   with   the   uniform 
density  p.     Then  it  can  be  shown  by  familiar  methods  that,  just  as 
in  the  classical  mechanics,  the  density  of  distribution  remains  uniform. 

Take,  for  example,  some  particular  cubical  element  of  our  gener- 
alized space  d4>i  dfa  dfo  •  -  -  d\f/i  d\I/%  dfa  •  •  • .  The  density  of  dis- 
tribution will  evidently  remain  uniform  if  the  number  of  points 
entering  any  such  cube  per  second  is  equal  to  the  number  leaving. 
Consider  now  the  two  parallel  bounding  surfaces  of  the  cube  which 
are  perpendicular  to  the  <£i  axis,  one  cutting  the  axis  at  the  point  fa 
and  the  other  at  the  point  <£i  +  d<f>i.  The  area  of  each  of  these 
surfaces  is  d  fad  fa  •  •  •  d\l/id\l/2d\l/3  •  •  •,  and  hence,  if  <j>i  is  the  component 

of  velocity  which  the  points  have  parallel  to  the  fa  axis,  and  -  -  is 

dfa 

the  rate  at  which  this  component  is  changing  as  we  move  along  the 
axis,  we  may  obviously  write  the  following  expression  for  the  differ- 
ence between  the  number  of  points  leaving  and  entering  per  second 
through  these  two  parallel  surfaces 


L/<£»1     %  I  »-'V>l 

~~~~  I  d<b\  I  rfd>2^^3  *  *  '  dwidu/^dy/s  •  •  •  =  p dV* 

d<{>i  /  d<pi 

Finally,  considering  all  the  pairs  of  parallel  bounding  surfaces,  we 
*  Jeans,  The  Dynamical  Theory  of  Gases,  Cambridge,  1916. 


104  Chapter  Eight. 

find  for  the  total  decrease  per  second  in  the  contents  of  the  element 

(d<i>i       602       60s  d\j/i       d\f/z    .    dfa  \ 

ZT  +  IT"  +  ~^T  +  '  '  '  '  +  TT  +  TT  +  TT  +  •  •  •  HF. 
601          602          ^03  6^1          6^2          d\l/S  J 

But  the  motions  of  the  points  are  necessarily  governed  by  the  Hamil- 
tonian  equations  (113)  given  above,  and  these  obviously  lead  to  the 
relations 

60i       d\I/i  _ 
60i  +  a^i  "     ' 


i        =  o 

602          6^2 

etc. 

So  that  our  expression  for  the  change  per  second  in  the  number  of 
points  in  the  cube  becomes  equal  to  zero,  the  necessary  requirement 
for  preserving  uniform  density. 

This  maintenance  of  a  uniform  distribution  means  that  there  is 
no  tendency  for  the  points  to  crowd  into  any  particular  region  of  the 
generalized  space,  and  hence  if  we  start  some  one  system  going  and 
plot  its  state  in  our  generalized  space,  we  may  assume  that,  after  an 
indefinite  lapse  of  time,  the  point  is  equally  likely  to  be  in  any  one  of 
the  little  elements  dV.  In  other  words,  the  different  states  of  a  system, 
which  we  can  specify  by  stating  the  region  dcfrid^zdfo  •  •  •  d\I/id\l/zd\j/3  •  •  • 
in  which  the  values  of  the  coordinates  and  momenta  of  the  system  fall, 
are  all  equally  likely  to  occur.* 

94.  A  System  of  Particles.  Consider  now  a  system  containing  Na 
particles  which  have  the  mass  ma  when  at  rest,  Nb  particles  which 
have  the  mass  nib,  Nc  particles  which  have  the  mass  mc,  etc.  If  at 
any  given  instant  we  specify  the  particular  differential  element 
dx  dy  dz  d^x  d$v  d\I/z  which  contains  the  coordinates  x,  y,  z,  and  the 
corresponding  momenta  \f/xt  \j/y,  \f/z  for  each  particle,  we  shall  thereby 
completely  determine  what  Planckf  has  well  called  the  microscopic 
state  of  the  system,  and  by  the  previous  paragraph  any  microscopic 

*  The  criterion  here  used  for  determining  whether  or  not  the  states  are  equally 
liable  to  occur  is  obviously  a  necessary  requirement,  although  it  is  not  so  evident 
that  it  is  a  sufficient  requirement  for  equal  probability. 

t  Planck,  Wdrmestrahlung,  Leipzig,  1913. 


Chaotic  Motion  of  a  System  of  Particles.  105 

state  of  the  system  in  which  we  thus  specify  the  six-dimensional 
position  of  each  particle  is  just  as  likely  to  occur  as  any  other  micro- 
scopic state. 

It  must  be  noticed,  however,  that  many  of  the  possible  micro- 
scopic states  which  are  determined  by  specifying  the  six-dimensional 
position  of  each  individual  particle  are  in  reality  completely  identical, 
since  if  all  the  particles  having  a  given  mass  ma  are  alike  among  them- 
selves, it  makes  no  difference  which  particular  one  of  the  various 
available  identical  particles  we  pick  out  to  put  into  a  specified  range 
dx  dy  dz  d^/x  d\j/v  d\j/z. 

For  this  reason  we  shall  usually  be  interested  in  specifying  the 
statistical  state*  of  the  system,  for  which  purpose  we  shall  merely 
state  the  number  of  particles  of  a  given  kind  which  have  coordinates 
falling  in  a  given  range  dx  dy  dz  d\f/x  d\f/y  d\f/z.  We  see  that  corre- 
sponding to  any  given  statistical  state  there  will  be  in  general  a 
large  number  of  microscopic  states. 

95.  Probability  of  a  Given  Statistical  State.  We  shall  now  be 
particularly  interested  in  the  probability  that  the  system  of  particles 
will  actually  be  in  some  specified  statistical  state,  and  since  Liou- 
ville's  theorem  has  justified  our  belief  that  all  microscopic  states  are 
equally  likely  to  occur,  we  see  that  the  probability  of  a  given  statis- 
tical state  will  be  proportional  to  the  number  of  microscopic  states 
which  correspond  to  it. 

For  the  system  under  consideration  let  a  particular  statistical 
state  be  specified  by  stating  that  Na',  Na",  Na'",  •  •  • ,  Nb',  Nb",  Nb"', 
•  •  • ,  etc.,  are  the  number  of  particles  of  the  corresponding  masses 
ma,  nib,  etc.,  which  fall  in  the  specified  elementary  regions  dx  dy  dz 
d\l/x  d\l/y  d\Jsz,  Nos.  la,  2a,  3a,  •••,  lb,  2b,  36,  ••"•,  etc.  By  familiar 
methods  of  calculation  it  is  evident  that  the  number  of  arrangements 
by  which  the  particular  distribution  of  particles  can  be  effected, 
that  is,  in  other  words,  the  number  of  microscopic  states,  W,  which 
correspond  to  the  given  statistical  state,  is  given  by  the  expression 

\Na\Nb\Nc... 


W  - 


\Ngf\Ng"    Ng'"   '  •  -  \Nb'\Nb"\Nb'" 


*  What  we  have  here  defined  as  the  statistical  state  is  what  Planck  calls  the 
macroscopic  state  of  the  system.  The  word  macroscopic  is  unfortunate,  however,  in 
implying  a  less  minute  observation  as  to  the  size  of  the  elements  dx  dy  dz  d\J/x  d\J/v  d\l/t 
in  which  the  representative  points  are  found. 


106  Chapter  Eight. 

and  this  number  W  is  proportional  to  the  probability  that  the  system 
will  be  found  in  the  particular  statistical  state  considered. 

If  now  we  assume  that  each  of  the  regions 

dx  dy  dz  d\f/x  d\f/y  oV*,  Nos.  la,  2a,  3a,  ••••,  16,  26,  36,  ••»,  etc. 
is  great  enough  to  contain  a  large  number  of  particles,*  we  may 
apply  the  Stirling  formula 


for  evaluating  |  JV0;  JVj,,  etc.,  and  omitting  negligible  terms,  shall 
obtain  for  log  W  the  result 

(AM          AM        AT"  "          AM'        AM"          N  '"  \ 

-flr  o    ,         *V  a     .,-•*•*••  i         -iV  a        .    -iV  a       ,         ^V  a         .  \ 

F:  log  w:  +  iv;  log  ^  +  TFT  log  iv-r  +  '  '  '  J 

.  (NS        Nb'      Nb"        Nb"      Nb'"        Nb'" 

-^l^^jvT+ivr^i^+ivr^^r+' 

etc. 

^Va'       Na" 

For  simplicity  let  us  denote  the  ratios  -^~-  ,  -r?-  ,  etc.,  by  the 

IV  a  -iV  o 

symbols  wa',  waft,  etc.  These  quantities  war,  wa",  etc.,  are  evidently 
the  probabilities,  in  the  case  of  this  particular  statistical  state, 
that  any  given  particle  ma  will  be  found  in  the  respective  regions 
Nos.  la,  2a,  etc. 

We  may  now  write 


log  W   =    —  NaZWa  log  Wa   ~  Nb2Wb  log  Wb   ~,    etc., 

where  the  summation  extends  over  all  the  regions  Nos.  la,  2a,  •  •  •  , 
16,  26,  etc. 

96.  Equilibrium  Relations.  Let  us  now  suppose  that  the  system 
of  particles  is  contained  in  an  enclosed  space  and  has  the  definite 
energy  content  E.  Let  us  find  the  most  probable  distribution  of  the 
particles.  For  this  the  necessary  condition  will  be 


5  log  W  =  -  N0S(log  wa  + 

-  N6S(log  wb  +  l)«u>6  •  •  •  =0.     (114) 

In  carrying  out  our  variation,  however,  the  number  of  particles  of 

*  The  idea  of  successive  orders  of  infinitesimals  which  permit  the  differential 
region  dx  dy  dz  d^x  d\f/y  dtyt  to  contain  a  large  number  of  particles  is  a  familiar  one  in 
mathematics. 


Chaotic  Motion  of  a  System  of  Particles.  107 

each  kind  must  remain  constant  so  that  we  have  the  added  relations 
25wa  =  0,         2dwb  =  0,         etc.  (115) 

Finally,  since  the  energy  is  to  have  a  definite  value  E,  it  must  also 
remain  constant  in  the  variation,  which  will  provide  still  a  further 
relation.  Since  the  energy  of  a  particle  will  be  a  definite  function  of 
its  position  and  momentum,*  let  us  write  the  energy  of  the  system 
in  the  form 

E    =   NaZwJBa  + 


where  Ea  is  the  energy  of  a  particle  in  the  region  la,  etc. 

Since  in  carrying  out  our  variation  the  energy  is  to  remain  con- 
stant, we  have  the  relation 

E  =  Na2Ea8Wa  +  Nb?Ebdwb  +  •  •  •  =0.  (116) 

Solving  the  simultaneous  equations  (114),  (115),  (116)  by  familiar 
methods  we  obtain 

log  wa  +  1  +  \Ea  +  M6  =  0, 

log  wb  +  1  +  \Eb  +  M6  =  0, 
etc., 

where  X,  /xa,  M&>  etc.,  are  undetermined  constants.  (It  should  be 
specially  noticed  that  X  is  the  same  constant  in  each  of  the  series  of 
equations.) 

Transforming  we  have 

Wa  =  aae~hEay 

wb  =  abe-hE>,  (117) 

etc., 

as  the  expressions  which  determine  the  chance  that  a  given  particle 
of  mass  ma,  mb,  etc.,  will  fall  in  a  given  region  dx  dy  dz  d\f/x  d\f/v  d\l/z, 
when  we  have  the  distribution  of  maximum  probability.  It  should 
be  noticed  that  h,  which  corresponds  to  the  X  of  the  preceding  equa- 
tions, is  the  same  constant  in  all  of  the  equations,  while  aa,  ab,  etc., 
are  different  constants,  depending  on  the  mass  of  the  particles  ma, 
mb)  etc. 

*  We  thus  exclude  from  our  considerations  systems  in  which  the  potential  energy 
depends  appreciably  on  the  relative  positions  of  the  independent  particles. 


108  Chapter  Eight. 

97.  The  Energy  as  a  Function  of  the  Momentum.  Ea,  Eb,  etc., 
are  of  course  functions  of  x,  y,  z,  \J/X,  \f/y,  ^2.  Let  us  now  obtain  an 
expression  for  Ea  in  terms  of  these  quantities.  If  there  is  no  external 
field  of  force  acting,  the  energy  of  a  particle  Ea  will  be  independent 
of  x,  y,  and  z,  and  will  be  determined  entirely  by  its  velocity  and 
mass.  In  accordance  with  the  theory  of  relativity  we  shall  have* 


(118) 


where  ma  is  the  mass  of  the  particle  at  rest. 

Let  us  now  express  Ea  as  a  function  of  \f/x,  \j/y,  \J/e. 

We  have  from  our  equations  (105)  and  (98),  which  were  used  for 
defining  momentum 


+  i 


max 


Constructing  the  similar  expressions  for  \I/y  and  \J/Z  we  may  write  the 
relation 


which  also  defines  \f/2. 

*  This  expression  is  that  for  the  total  energy  of  the  particle,  including  that 
internal  energy  w0c2  which,  according  to  relativity  theory,  the  particle  has  when 
it  is  at  rest.  (See  Section  75.)  It  would  be  just  as  correct  to  substitute  for  Ea  in 

equation  (117)  the  value  of  the  kinetic  energy  mac2  /  —  --  —  1  \  instead  of  the 


7W    /> 

total  energy  —    ffl        ,  since  the  two  differ  merely  by  a  constant  mac2  which  would 
/i       u* 

\l~j 

be  taken  care  of  by  assigning  a  suitable  value  to  aa. 


Chaotic  Motion  of  a  System  of  Particles.  109 

By  simple  transformations  and  the  introduction  of  equation  (118) 
we  obtain  the  desired  relation 


Ea  =  c    ^2  +  ma2c2.  (120) 

98.  The  Distribution  Law.     We  may  now  rewrite  equations  (117) 
in  the  form 


wa  =  a 

wb  =  abe-he"^+m^t  (121) 

etc. 

These  expressions  determine  the  probability  that  a  given  particle 
of  mass  ma)  m^  etc.  will  fall  in  a  given  region  dxdydzd\f/xd\f/yd\f/Z)  and 
correspond  to  Maxwell's  distribution  law  in  ordinary  mechanics.  We 
see  that  these  probabilities  are  independent  of  the  position  x,  y}  z* 
but  dependent  on  the  momentum. 

aae'*******1"***  is  the  probability  that  a  given  particle  will  fall  in  a 
particular  six-dimensional  cube  of  volume  dxdydzd\j/xd\J/yd\f/z.  Let  us 
now  introduce,  for  convenience,  a  new  quantity  aae~hc  *f+m-2ci  which 
will  be  the  probability  per  unit  volume  that  a  given  particle  will  have 
the  six  dimensional  location  in  question,  the  constants  aa  and  aa 
standing  in  the  same  ratio  as  the  volumes  dxdydzd\l/xd\l/yd\l/z  and  unity. 

We  may  then  write 


wa  = 
wb  = 


etc. 


Since  every  particle  must  have  components  of  momentum  lying 
between  minus  and  plus  infinity,  and  lie  somewhere  in  the  whole 
volume  V  occupied  by  the  mixture,  we  have  the  relation 


£00        /»+00        /»+ 
)       «/  —  00       J—  00 


=  1.  (122) 


It  is  further  evident  that  the  average  value  of  any  quantity  A 
which  depends  on  the  momentum  of  the  particles  is  given  by  the 
*  This  is  true  only  when,  as  assumed,  no  external  field  of  force  is  acting. 


110  Chapter  Eight. 

expression 

/»+w      /»+«      /»+«  __ 

[ALv.  =  71  aae-hc^m"*c*Adtxdtydtz,      (123) 

t/_00        1/-00        */-00 

where  A  is  some  function  of  $x,  $v,  and  ^2. 

99.  Polar  Coordinates.  We  may  express  relations  corresponding 
to  (122)  and  (123)  more  simply  if  we  make  use  of  polar  coordinates. 
Consider  instead  of  the  elementary  volume  d\f/xd\l/vd\f/z  the  volume 
\l/2  smdddd4>d\f/  expressed  in  polar  coordinates,  where 


The    probability   that    a   particle   ma   will    fall    in    the   region 
dxdydz\f/2  smdddd<f>d\f/  will  be 


sin 

and  since  each  particle  must  fall  somewhere  in  the  space  x  y  z  \f/x  \l/l 
we  shall  have  corresponding  to  (122)  the  relation 

X7T         /»2lT         /»00 
a«, 
c/o         ty  0 

(124) 

-tcJ^j*  ,2d,  =  L 


Jo 


Corresponding  to  equation  (123),  we  also  see  that  the  average  value 
of  any  quantity  A,  which  is  dependent  on  the  momentum  of  the 
molecules  of  mass  ma,  will  be  given  by  the  expression 


[A]av  =47r7        aae~hc^z+m^  A^d^.  (125) 

Jo 

100.  The  Law  of  Equipartition.  We  may  now  obtain  a  law  which 
corresponds  to  that  of  the  equipartition  of  vis  viva  in  the  classical 
mechanics.  Considering  equation  (124)  let  us  integrate  by  parts,  we 
obtain 


_!«/,=<) 

r^aae-h^*^^(-  he) 
Jo     3 


Chaotic  Motion  of  a  System  of  Particles.  Ill 

Substituting  the  limits  into  the  first  term  we  find  that  it  becomes 
zero  and  may  write 


a  , 

o  V^2  +  m02c2 

But  by  equation  (125)  the  left-hand  side  of  this  relation  is  the 

\I/2c 

average  value  of  -  =  for  the  particles  of  mass  ma.     We  have 

m02c2 


Introducing  equation  (119)  which  defines  i/'2,  we  may  transform  this 
expression  into 


mau2 


I  <126> 


Since  we  have  shown  that  h  is  independent  of  the  mass  of  the 

m,QU2 
particles,  we  see  that  the  average  value  of  — .—          is  the  same  for  particles 


of  all  different  masses.  This  is  the  principle  in  relativity  mechanics 
that  corresponds  to  the  law  of  the  equipartition  of  vis  viva  in  the 
classical  mechanics.  Indeed,  for  low  velocities  the  above  expression 
reduces  to  m0w2,  the  vis  viva  of  Newtonian  mechanics,  a  fact  which 
affords  an  illustration  of  the  general  principle  that  the  laws  of  New- 
tonian mechanics  are  always  the  limiting  form  assumed  at  low  veloci- 
ties by  the  more  exact  formulations  of  relativity  mechanics. 

We  may  now  call  attention  in  passing  to  the  fact  that  this  quantity 


,  whose  value  is  the  same  for  particles  of  different  masses,  is 


not  the  relativity  expression  for  kinetic  energy,  which  is  given  rather 

by  the  formula  c2  I         ^=  —  mQ    .     So  that  in  relativity  mechanics 

'1  -  — 
1       c2 


112  Chapter  Eight. 

the  principle  of  the  equipartition  of  energy  is  merely  an  approximation. 
We  shall  later  return  to  this  subject. 

101.  Criterion  for  Equality  of  Temperature.  For  a  system  of  par- 
ticles of  masses  ma,  w&,  etc.,  enclosed  in  the  volume  Vt  and  having  the 
definite  energy  content  E,  we  have  shown  that 


and 


are  the  respective  probabilities  that  given  particles  of  mass  ma  or 
mass  nib  will  have  momenta  between  \I/  and  ty  +  d\l/.  Suppose  now 
we  consider  a  differently  arranged  system  in  which  we  have  Na  par- 
ticles of  mass  ma  by  themselves  in  a  space  of  volume  Va  and  Ni 
particles  of  mass  ra&  in  a  contiguous  space  of  volume  F&,  separated 
from  Va  by  a  partition  which  permits  a  transfer  of  energy,  and  let 
the  total  energy  of  the  double  system  be,  as  before,  a  definite  quantity 
E  (the  energy  content  of  the  partition  being  taken  as  negligible). 
Then,  by  reasoning  entirely  similar  to  that  just  employed,  we  can 
obviously  show  that 

and 


are  now  the  respective  probabilities  that  given  particles  of  mass  ma 
or  mass  m&  will  have  momenta  between  ^  and  \f/  -f  d\f/,  the  only 
changes  in  the  expressions  being  the  substitution  of  the  volumes 
Va  and  Vb  in  the  place  of  the  one  volume  V.  Furthermore,  this 
distribution  law  will  evidently  lead  as  before  to  the  equality  of  the 

average  values  of 

mav? 

and 


!i 

72 

Since,  however,  the  spaces  containing  the  two  kinds  of  particles  are  in 
thermal  contact,  their  temperature  is  the  same.     Hence  we  find  that 

the  equality  of  the  average  values  of  —j=  =====  is  the  necessary  condition  for 

equality  of  temperature. 


Chaotic  Motion  of  a  System  of  Particles.  113 

The  above  distribution  law  also  leads  to  the  important  corollary  that 
for  any  given  system  of  particles  at  a  definite  temperature  the  momenta 
and  hence  the  total  energy  content  is  independent  of  the  volume. 

We  may  now  proceed  to  the  derivation  of  relations  which  will 


permit  us  to  show  that  the  important  quantity      .  --  is  directly 


proportional  to  the  temperature  as  measured  on  the  absolute  ther- 
modynamic  temperature  scale. 

102.  Pressure  Exerted  by  a  System  of  Particles.  We  first  need 
to  obtain  an  expression  for  the  pressure  exerted  by  a  system  of  N 
particles  enclosed  in  the  volume  V.  Consider  an  element  of  surface 
dS  perpendicular  to  the  X  axis,  and  let  the  pressure  acting  on  it  be  p. 
The  total  force  which  the  element  dS  exerts  on  the  particles  that 
impinge  will  be  pdS,  and  this  will  be  equal  to  the  rate  of  change  of 
the  momenta  in  the  X  direction  of  these  particles.* 

Now  by  equation  (122)  the  total  number  of  particles  having 
momenta  between  \f/x  and  \J/X  +  d\I/x  in  the  positive  direction  is 

f*$x+d\lif     /»+« 

NV 


,   /•+*    /•+ 

J—  oo      J— oo 


But  xdS  gives  us  the  volume  which  contains  the  number  of  particles 
having  momenta  between  \J/X  and  \f/x  +  d\f/x  which  will  reach  dS  in  a 
second.  Hence  the  number  of  such  particles  which  impinge  per 
second  will  be 


xds  r**+d**  r 

-yr  I 

V     J  J_ 


and  their  change  in  momentum,  allowing  for  the  effect  of  the  rebound, 
will  be 


z+Wg       /»+»       /» 

J—n     J— 


^, 

Finally,  the  total  change  in  momentum  per  second  for  all  particles 
can  be  found  by  integrating  for  all  possible  positive  values  of  \f/x. 

*  The  system  is  considered  dilute  enough  for  the  mutual  attractions  of  the 
particles  to  be  negligible  hi  their  effect  on  the  external  pressure. 
9 


114  Chapter  Eight. 

Equating  this  to  the  total  force  pdS  we  have 
pdS  =  2NdS  r  f    °  f+°°ae-Acl 

Jo       J-oo     J-» 

Cancelling  dS,  multiplying  both  sides  of  the  equation  by  the  volume  7, 
changing  the  limits  of  integration  and  substituting      m0x       for  \f/x, 


we  have 

•+00 


/•+«        /•-fW         y'.+ 

l  I 

*/_oo      fc/-oo      */  — oo 


But  this  by  equation  (123)  reduces  to 
pV  = 


or,  since 

mnu2  mv_.  ...vy 

1  i      / ~   i 


we  have  from  symmetry 


Since  at  a  given  temperature  we  have  seen  that  the  term  in  parenthesis 
is  independent  of  the  volume  and  the  nature  of  the  particles,  we  see 
that  the  laws  of  Boyle  and  Avogadro  hold  also  in  relativity  mechanics 
for  a  system  of  particles. 

For  slow  velocities  equation  (127)  reduces  to  the  familiar  expression 

N 
pV  =  —  (m0w2)av.. 

103.  The  Relativity  Expression  for  Temperature.  We  are  now  in 
a  position  to  derive  the  relativity  expression  for  temperature.  The 
thermodynamic  scale  of  temperature  may  be  defined  in  terms  of  the 
efficiency  of  a  heat  engine.  Consider  a  four-step  cycle  performed 
with  a  working  substance  contained  in  a  cylinder  provided  with  a 
piston.  In  the  first  step  let  the  substance  expand  isothermally  and 


Chaotic  Motion  of  a  System  of  Particles.  115 

reversibly,  absorbing  the  heat  Q2  from  a  reservoir  at  temperature  T2; 
in  the  second  step  cool  the  cylinder  down  at  constant  volume  to  T\] 
in  the  third  step  compress  to  the  original  volume,  giving  out  the 
heat  Qi  at  temperature  TI,  and  in  the  fourth  step  heat  to  the  original 
temperature.  Now  if  the  working  substance  is  of  such  a  nature  that 
the  heat  given  out  in  the  second  step  could  be  used  for  the  reversible 
heating  of  the  cylinder  in  the  fourth  step,  we  may  define  the  absolute 

T        O 

temperatures  T2  and  T\  by  the  relation  —  =  -r-  .* 

1 1      ^i 

Consider  now  such  a  cycle  performed  on  a  cylinder  which  con- 
tains one  of  our  systems  of  particles.  Since  we  have  shown  (Section 
101)  that  at  a  definite  temperature  the  energy  content  of  such  a 
system  is  independent  of  the  volume,  it  is  evident  that  our  working 
substance  fulfils  the  requirement  that  the  heat  given  out  in  the  second 
step  shall  be  sufficient  for  the  reversible  heating  in  the  last  step. 
Hence,  in  accordance  with  the  thermodynamic  scale,  we  may  measure 

m  f\ 

the  temperatures  of  the  two  heat  reservoirs  by  the  relation  -=?  =  ~ 

J- 1      <Ji 
and  may  proceed  to  obtain  expressions  for  Q2  and  Qi. 

In  order  to  obtain  these  expressions  we  may  again  make  use  of  the 
principle  that  the  energy  content  at  a  definite  temperature  is  inde- 
pendent of  the  volume.  This  being  true,  we  see  that  Q2  and  Qi 
must  be  equal  to  the  work  done  in  the  changes  of  volume  that  take 
place  respectively  at  Tz  and  TI,  and  we  may  write  the  relations 


Q2=    CV 

Jv 

Qi  =    f    pdV(&t  T,). 
Jr 


v 

But  equation  (127)  provides  an  expression  for  p  in  terms  of  V,  leading 
on  integration  to  the  relations 


*  We  have  used  this  cycle  for  defining  the  thermodynamic  temperature  scale 
instead  of  the  familiar  Carnot  cycle,  since  it  avoids  the  necessity  of  obtaining  an 
expression  for  the  relation  between  pressure  and  volume  in  an  adiabatic  expansion. 


116 


Chapter  Eight. 


N 

Oi-j 


which  gives  us  on  division 


Ui1 


We  see  £/ia£  ZAe  absolute  temperature  measured  on  the  thermodynamic 
scale  is  proportional  to  the  average  value  af 


We  may  finally  express  our  temperature  in  the  same  units  custom- 
arily employed  by  comparing  equation  (127) 

,r-f 


with  the  ordinary  form  of  the  gas  law 

.     pV  =  nRT, 

where  n  is  the  number  of  mols  of  gas  present. 
We  evidently  obtain 

m0u2 


•i-S 

c2  J 


N          m0u 


3nR 


(128) 


V  J  av.  L    \  C*  J  av. 

where  the  quantity  — ,  which  may  be  called  the  gas  constant  for  a 
single  molecule,  has  been  denoted,  as 'is  customary,  by  the  letter  k. 


Chaotic  Motion  of  a  System  of  Particles.  117 

Remembering  the  relation  =  T  ,  we  have 


(129) 

104.  The  Partition  of  Energy.  We  have  seen  that  our  new  equi- 
partition  law  precludes  the  possibility  of  an  exact  equipartition  of 
energy.  It  becomes  very  important  to  see  what  the  average  energy 
of  a  particle  of  a  given  mass  does  become  at  any  temperature. 

Equation  (125)  provides  a  general  expression  for  the  average  value 
of  any  property  of  the  particles.  For  the  average  value  of  the  energy 
-f  ?rc02c2  of  particles  of  mass  m0  (see  equation  120)  we  shall  have 


[E]&v  =  4x7  f   ae-^^V'cV^2  + 
Jo 


The  unknown  constant  a  may  be  eliminated  with  the  help  of  the 
relation  (124) 

4x7  I     ae-hci/+*+"tfc*  ^d^  =  1 
Jo 

and  for  h  we  may  substitute  the  value  given  by  (129),  which  gives  us 
the  desired  equation 


1 


I        e- 

Jo 


(130) 


105.  Partition  of  Energy  for  Zero  Mass.  Unfortunately,  no  gen- 
eral method  for  the  evaluation  of  this  expression  seems  to  be  available. 
For  the  particular  case  that  the  mass  m0  of  the  particles  approaches 
zero  compared  to  the  momentum,  the  expression  reduces  to 


f    c- 

° 


,-(c^/*r) 


118  Chapter  Eight. 

in  terms  of  integrals  whose  values  are  known.     Evaluating,  we  obtain 

[£]„.  =  3kT. 

For  the  total  energy  of  N  such  particles  we  obtain 

E  =--  3NkT, 

and  introducing  the  relation  k  =  —  by  which  we  defined  k  we  have 

E  =  ZnRT  (131) 

as  the  expression  for  the  energy  of  n  mols  of  particles  if  their  value  of 
mo  is  small  compared  with  their  momentum. 

It  is  instructive  to  compare  this  with  the  ordinary  expression  of 
Newtonian  mechanics 

E  =  \nRT, 

which  undoubtedly  holds  when  the  masses  are  so  large  and  the  veloci- 
ties so  small  that  no  appreciable  deviations  from  the  laws  of  New- 
tonian mechanics  are  to  be  expected.  We  see  that  for  particles  of 
very  small  mass  the  average  kinetic  energy  at  any  temperature  is 
twice  as  large  as  that  for  large  particles  at  the  same  temperature. 
It  is  also  interesting  to  note  that  in  accordance  with  equation  (131) 
a  mol  of  particles  which  approach  zero  mass  at  the  absolute  zero, 
would  have  a  mass  of 

3  X  8.31  XW  X  300  m  7  ^  x  iQ_u 

grams  at  room  temperature  (300°  absolute).  This  suggests  a  field 
of  fascinating  if  profitless  speculation. 

106.  Approximate  Partition  of  Energy  for  Particles  of  any  Desired 
Mass.  For  particles  of  any  desired  mass  we  may  obtain  an  approxi- 
mate idea  of  the  relation  between  energy  and  temperature  by  ex- 
panding the  expression  for  kinetic  energy  into  a  series.  For  the  aver- 
age kinetic  energy  of  a  particle  we  have 


-  mo 


Chaotic  Motion  of  a  System  of  Particles.  119 

Expanding  into  a  series  we  obtain  for  the  total  kinetic  energy  of  N 
particles 


where  u2,  u4,  etc.,  are  the  average  values  of  v?,  u*,  etc.,  for  the  indi- 
vidual particles. 

To  determine  approximately  how  the  value  of  K  varies  with  the 
temperature  we  may  also  expand  our  expression  (128)  for  temperature, 


into  a  series;  we  obtain 

1  u4       3   u6      15 


.     (133) 


Combining  expressions   (132)  and  (133)   by  subtraction  and  trans- 
position, we  obtain 


For  the  case  of  velocities  low  enough  so  that  u4  and  higher  powers 

can  be  neglected,  this  reduces  to  the  familiar  expression  of  Newtonian 

3 
mechanics,  K  =  -  nRT. 

In  case  we  neglect  in  expression  (134)  powers  higher  than  u4  we 
have  the  approximate  relation 


8c2          2NmoC2  \      2 

the  left-hand  term  really  being  the  larger,  since  the  average  square  of  a 
quantity  is  greater  than  the  square  of  its  average.     Since  (  — ~—  ) 

/3          \2 
is  approximately  equal  to  I  =  nRT  1  ,  we  may  write  the  approxima- 


120  Chapter  Eight. 

tion 


or,  noting  that  NmQ  =  M,  the  total  mass  of  the  system  at  the  abso- 
lute zero,  we  have 


If  we  use  the  erg  as  our  unit  of  energy,  R  will  be  8.31  X  107;  expressing 
velocities  in  centimeters  per  second,  c2  will  be  1021,  and  M  will  be  the 
mass  of  the  system  in  grams. 

For  one  mol  of  a  monatomic  gas  we  should  have  in  ergs 

K  =  12.4  X  ItfT  +  1~  10-6T2. 

In  the  case  of  the  electron  M  may  be  taken  as  approximately 
1/1800.  At  room  temperature  the  second  term  of  our  equation  would 
be  entirely  negligible,  being  only  3.5  X  10~6  per  cent  of  the  first,  and 
still  be  only  3.5  X  10"4  per  cent  in  a  fixed  star  having  a  temperature  of 
30,000°.  Hence  at  all  ordinary  temperatures  we  may  expect  the 
law  of  the  equipartition  of  energy  to  be  substantially  exact  for  par- 
ticles of  mass  as  small  as  the  electron. 

Our  purpose  in  carrying  through  the  calculations  of  this  chapter 
has  been  to  show  that  a  very  important  and  interesting  problem  in 
the  classical  mechanics  can  be  handled  just  as  easily  in  the  newer 
mechanics,  and  also  to  point  out  the  nature  of  the  modifications  in 
existing  theory  which  will  have  to  be  introduced  if  the  later  develop- 
ments of  physics  should  force  us  to  consider  equilibrium  relations  for 
particles  of  mass  much  smaller  than  that  of  the  electron. 

We  may  also  call  attention  to  the  fact  that  we  have  here  con- 
sidered a  system  whose  equations  of  motion  agree  with  the  principles 
of  dynamics  and  yet  do  not  lead  to  the  equipartition  of  energy.  This 
is  of  particular  interest  at  a  time  when  many  scientists  have  thought 
that  the  failure  of  equipartition  in  the  hohlraum  stood  in  necessary 
conflict  with  the  principles  of  dynamics. 


CHAPTER  IX. 

THE  PRINCIPLE  OF  RELATIVITY  AND  THE  PRINCIPLE  OF 
LEAST  ACTION. 

It  has  been  shown  by  the  work  of  Helmholtz,  J.  J.  Thomson, 
Planck  and  others  that  the  principle  of  least  action  is  applicable  in 
the  most  diverse  fields  of  physical  science,  and  is  perhaps  the  most 
general  dynamical  principle  at  our  disposal.  Indeed,  for  any  system 
whose  future  behavior  is  determined  by  the  instantaneous  values  of  a 
number  of  coordinates  and  their  time  rate  of  change,  it  seems  possible 
to  throw  the  equations  describing  the  behavior  of  the  system  into 
the  form  prescribed  by  the  principle  of  least  action.  This  generality 
of  the  principle  of  least  action  makes  it  very  desirable  to  develop  the 
relation  between  it  and  the  principle  of  relativity,  and  we  shall  obtain 
in  this  way  the  most  important  and  most  general  method  for  deriving 
the  consequences  of  the  theory  of  relativity.  We  have  already 
developed  in  Chapter  VII  the  particular  application  of  the  principle 
of  least  action  in  the  case  of  a  system  of  particles,  and  with  the  help 
of  the  more  general  development  which  we  are  about  to  present,  we 
shall  be  able  to  apply  the  principle  of  relativity  to  the  theories  of 
elasticity,  of  thermodynamics  and  of  electricity  and  magnetism. 

107.  The  Principle  of  Least  Action.  For  our  purposes  the  prin- 
ciple of  least  action  may  be  most  simply  stated  by  the  equation 

=  0.  (135) 

This  equation  applies  to  any  system  whose  behavior  is  determined 
by  the  values  of  a  number  of  independent  coordinates  fafafa  -  •  • 
and  their  rate  of  change  with  the  time  (frifafo  •  •  •,  and  the  equation 
describes  the  path  by  which  the  system  travels  from  its  configuration 
at  any  time  ti  to  its  configuration  at  any  subsequent  time  tz. 

H  is  the  so-called  kinetic  potential  of  the  system  and  is  a  func- 
tion of  the  coordinates  and  their  generalized  velocities  : 


H  =  F(0i020,  •  •  •  <£i<«3  -  •  •)•  (136) 

121 


122  Chapter  Nine. 

$H  is  the  variation  of  H  at  any  instant  corresponding  to  a  slightly 
displaced  path  by  which  the  system  might  travel  from  the  same 
initial  to  the  same  final  state  in  the  same  time  interval,  and  W  is  the 
external  work  corresponding  to  the  variation  5  which  would  be  done 
on  the  system  by  the  external  forces  if  at  the  instant  in  question  the 
system  should  be  displaced  from  its  actual  configuration  to  its  con- 
figuration on  the  displaced  path.  Thus 

W  =  <M<£i  +  $2S02  +  *s«08  +.'••,  (137) 

where  $1,  $2,  etc.,  are  the  so-called  generalized  external  forces  which 
act  in  such  a  direction  as  to  increase  the  values  of  the  corresponding 
coordinates. 

The  form  of  the  function  which  determines  the  kinetic  potential 
H  depends  on  the  particular  nature  of  the  system  to  which  the  principle 
of  least  action  is  being  applied,  and  it  is  one  of  the  chief  tasks  of 
general  physics  to  discover  the  form  of  the  function  in  the  various 
fields  of  mechanical,  electrical  and  thermodynamic  investigation. 
As  soon  as  we  have  found  out  experimentally  what  the  form  of  H  is 
for  any  particular  field  of  investigation,  the  principle  of  least  action, 
as  expressed  by  equation  (135),  becomes  the  basic  equation  for  the 
mathematical  development  of  the  field  in  question,  a  development 
which  can  then  be  carried  out  by  well-known  methods. 

The  special  task  for  the  theory  of  relativity  will  be  to  find  a  general 
relation  applicable  to  any  kind  of  a  system,  which  shall  connect  the 
value  of  the  kinetic  potential  H  as  measured  with  respect  to  a  set  of 
coordinates  S  with  its  value  H'  as  measured  with  reference  to  another 
set  of  coordinates  Sf  which  is  in  motion  relative  to  S.  This  relation 
will  of  course  be  of  such  a  nature  as  to  agree  with  the  principle  of  the 
relativity  of  motion,  and  in  this  way  we  shall  introduce  the  principle 
of  relativity  at  the  very  start  into  the  fundamental  equation  for  all 
fields  of  dynamics. 

Before  proceeding  to  the  solution  of  that  problem  we  may  put 
the  principle  of  least  action  into  another  form  which  is  sometimes 
more  convenient,  by  obtaining  the  equations  for  the  motion  of  a 
system  in  the  so-called  Lagrangian  form. 

108.  The  Equations  of  Motion  in  the  Lagrangian  Form.  To  ob- 
tain the  equations  of  motion  in  the  Lagrangian  form  we  may  evidently 


Relativity  and  the  Principle  of  Least  Action.  123 

rewrite  our  fundamental  equation  (135)  in  the  form 
H  BH  dH  dH 


=  0. 
We  have  now,  however, 

d  d 


,        etc., 
which  gives  us 


or,  since  50i,  602,  etc.,  are  by  hypothesis  zero  at  times  ii  and  £2,  we 
obtain 


etc. 
On  substituting  these  expressions  in  (138)  we  obtain 


fKi-Ki)  +'. 

+(£-$(£)+»)•*+.-]•- 

arid  since  the  variations  of  0i,  02,  etc.,  are  entirely  independent  and 
the  limits  of  integration  ti  and  tz  are  entirely  at  our  disposal,  this 
equation  will  be  true  only  when  each  of  the  following  equations  is 
true.  And  these  are  the  equations  of  motion  in  the  desired  Lagrangian 


124  Chapter  Nine. 

form, 


(139) 


etc. 


In  these  equations  H  is  the  kinetic  potential  of  a  system  whose 
state  is  determined  by  the  generalized  coordinates  $1,  <£2,  etc.,  and 
their  time  derivatives  <£i,  $2,  etc.,  where  $1,  $2,  etc.,  are  the  gener- 
alized external  forces  acting  on  the  system  in  such  a  sense  as  to  tend 
to  increase  the  values  of  the  corresponding  generalized  coordinates. 

109.  Introduction  of  the  Principle  of  Relativity.  Let  us  now  in- 
vestigate the  relation  between  our  dynamical  principle  and  the  prin- 
ciple of  the  relativity  of  motion.  To  do  this  we  must  derive  an  equa- 
tion for  transforming  the  kinetic  potential  H  for  a  given  system 
from  one  set  of  coordinates  to  another.  In  other  words,  if  S  and  S' 
are  two  sets  of  reference  axes,  Sf  moving  past  S  in  the  X-direction 
with  the  velocity  V,  what  will  be  the  relation  between  H  and  H', 
the  values  for  the  kinetic  potential  of  a  given  system  as  measured 
with  reference  to  S  and  S"! 

It  is  evident  from  the  theory  of  relativity  that  our  fundamental 
equation  (135)  must  hold  for  the  behavior  of  a  given  system  using 
either  set  of  coordinates  S  or  S',  so  that  both  of  the  equations 

(dH  +  W)dt  =  0        and          f     (dH'  +  W')dtf  =  0    (140) 

Jt'i' 

or 

f  2  (dH  +  W}dt  =    f  2  (dHf  +  W')dt'  =  0 

J<!  Jtjf 

must  hold  for  a  given  process,  where  it  will  be  necessary,  of  course, 
to  choose  the  limits  of  integration  £1  and  t2,  t\  and  tz'  wide  enough 
apart  so  that  for  both  sets  of  coordinates  the  varied  motion  will  be 
completed  within  the  time  interval.  Since  we  shall  find  it  possible 

now  to  show  that  in  general  J  Wdt  =  J  W'dt',  we  shall  be  able  to 

obtain  from  the  above  equations  a  simple  relation  between  H  and  H'  '. 

1  10.  Relation  between  /  Wrdt'  and  /  Wdt.     To  obtain  the  desired 


Relativity  and  the  Principle  of  Least  Action.  125 

proof  we  must  call  attention  in  the  first  place  to  the  fact  that  all 
kinds  of  force  which  can  act  at  a  given  point  must  be  governed  by 
the  same  transformation  equations  when  changing  from  system  £  to 
system  S'.  This  arises  because  when  two  forces  of  a  different  nature 
are  of  such  a  magnitude  as  to  exactly  balance  each  other  and  produce 
no  acceleration  for  measurements  made  with  one  set  of  coordinates 
they  must  evidently  do  so  for  any  set  of  coordinates  (see  Chapter  IV, 
Section  42).  Since  we  have  already  found  transformation  equations 
for  the  force  acting  at  a  point,  in  our  consideration  of  the  dynamics 
of  a  particle,  we  may  now  use  these  expressions  in  general  for  the 

evaluation  /  W  'dtf. 

W  is  the  work  which  would  be  done  by  the  external  forces  if  at 
any  instant  t'  we  should  displace  our  system  from  its  actual  con- 
figuration to  the  simultaneous  configuration  on  the  displaced  path. 

Hence  it  is  evident  that  J  W'dt  will  be  equal  to  a  sum  of  terms  of  the 
type 

/  (Fx'dxf  +  Fy'dS  +  Fz'BW, 

where  Fzr,  Fy',  Fz',  is  the  force  acting  at  a  given  point  of  the  system 
and  8x',  5y',  5z'  are  the  displacements  necessary  to  reach  the  corre- 
sponding point  on  the  displaced  path,  all  these  quantities  being 
measured  with  respect  to  S'. 

Into  this  expression  we  may  substitute,  however,  in  accordance 
with  equations  (61),  (62),  (63)  and  (13),  the  values 


_ 

"  c2 


1_*F 


(141) 


126  Chapter  Nine. 

We  may  also  make  substitutions  for  dxf ',  5y'  and  dz'  in  terms  of 
dx,  by  and  dz,  but  to  obtain  transformation  equations  for  these  quanti- 
ties is  somewhat  complicated  owing  to  the  fact  that  positions  on  the 
actual  and  displaced  path,  which  are  simultaneous  when  measured 
with  respect  to  Sf,  will  not  be  simultaneous  with  respect  to  S.  We 
have  denoted  by  tf  the  time  in  system  S'  when  the  point  on  the  actual 
path  has  the  position  x',  y',  z'  and  simultaneously  the  point  on  the 
displaced  path  has  the  position  (xf  +  8x'),  (yf  +  5yf),  (zf  +  dz'), 
when  measured  in  system  S',  or  by  our  fundamental  transformation 
equations  (9),  (10)  and  (11)  the  positions  K(X'  +  Vtf),  y',  z'  and 
K([X'  +  dx']  +  Vt'),  (yr  +  dy'),  (zf  +  dz')  when  measured  in  system  S. 
If  now  we  denote  by  tA  and  tD  the  corresponding  times  in  system  S 
we  shall  have,  by  our  fundamental  transformation  equation  (12), 


and  we  see  that  in  system  S  the  point  has  reached  the  displaced 
position  at  a  time  later  than  that  of  the  actual  position  by  the  amount 

Ky 
t»-tA=  —  8x'  , 

and,  since  during  this  time-interval  the  displaced  point  would  have 
moved,  neglecting  higher-order  terms,  the  distances 

KV  KV  KV 


these  quantities  must  be  subtracted  from  the  coordinates  of  the 
displaced  point  in  order  to  obtain  a  position  on  the  displaced  path 
which  will  be  simultaneous  with  tA  as  measured  in  system  S.  We 
obtain  for  the  simultaneous  position  on  the  displaced  path 

xV  iiV 

K([xf  +  dx']  +  Vt')  -  K  —  5x',        y'  +  Sy'  -  K  ^  x', 

c  c 


Relativity  and  the  Principle  of  Least  Action. 
and  for  the  corresponding  position  on  the  actual  path 


127 


y'f 


and  obtain  by  subtraction 


=  V  -  «      -  to', 


=  ««'  -  *  —  to'. 
c2 


(142) 


Substituting  now  these  equations,   together  with  the  other  trans- 
formation equations  (141),  in  our  expression  we  obtain 


Vto'  +  Fy'6y'  +  F.'Sz'W 


yV     F,          zV     F, 

f  x    —    — IT 


xV       c2 


xV 


xV 


8x 


(143) 


-  -r  }dt 


=  f  (Fz8x  +  Fvdy  +  1 
We  thus  see  that  we  must  always  have  the  general  equality 

/  W'dt'  =  f  Wdt. 


(144) 


111.  Relation  between  H'  and  H.  Introducing  this  equation  into 
our  earlier  expression  (140)  we  obtain  as  a  general  relation  between 
H'  and  H 

f  8H'dt'  =  /  dHdt.  (145) 

Restricting  ourselves  to  systems  of  such  a  nature  that  we  can 


128  Chapter  Nine. 

assign  them  a  definite  velocity  u  =  xi  +  yj  -f  zk,  we  can  rewrite 
this  expression  in  the  following  form,  where  by  HD  and  HA  we  denote 
the  values  of  the  kinetic  potential  respectively  on  the  displaced  and 
actual  paths 

J  tHW  =  J  H»'dt'  -   f  HA'dtf  =  J  HJ*  (  1  -  ^y—  )  dt 

-  J#A  (l  -  ^)  (ft  =  f  Hvdt  -  J  JG^ctt, 
and  hence  obtain  for  such  systems  the  simple  expression 


/       t?2         /       w* 

Noting  the  relation  between  \l  1  —  —  and  -v/  1  -  —  given  in  equation 

»  c  '  c 


c1 
(17),  this  can  be  rewritten 

H'  H 


(146) 


and  this  is  the  expression  which  we  shall  find  most  useful  for  our 
future  development  of  the  consequences  of  the  theory  of  relativity. 
Expressing  the  requirement  of  the  equation  in  words  we  may  say 

TT 

that  the  theory  of  relativity  requires  an  invariance  of  —  .-  in  the 


Lorentz  transformation. 

112.  As  indicated  above,  the  use  of  this  equation  is  obviously 
restricted  to  systems  moving  with  some  perfectly  definite  velocity  u. 
Systems  satisfying  this  condition  would  include  particles,  infinitesimal 
portions  of  continuous  systems,  and  larger  systems  in  a  steady  state. 

113.  Our  general  method  of  procedure  in  different  fields  of  investi- 
gation will  now  be  to  examine  the  expression  for  kinetic  potential 
which  is  known  to  hold  for  the  field  in  question,  provided  the  velocities 
involved  are  low  and  by  making  slight  alterations  when  necessary, 


Relativity  and  the  Principle  of  Least  Action.  129 

see  if  this  expression  can  be  made  to  agree  with  the  requirements  of 
equation  (146)  without  changing  its  value  for  low  velocities.  Thus 
it  is  well  known,  for  example,  that,  in  the  case  of  low  velocities,  for  a 
single  particle  acted  on  by  external  forces  the  kinetic  potential  may 
be  taken  as  the  kinetic  energy  Jm0w2.  For  relativity  mechanics,  as 
will  be  seen  from  the  developments  of  Chapter  VII,  we  may  take  for 

/         U? 
the  kinetic  potential,  —  Woe2  -y  1  -  -  —  ,  an  expression  which,  except  for 

an  additive  constant,  becomes  identical  with  \mtftf  at  low  velocities, 
and  which  at  all  velocities  agrees  with  equation  (146). 


10 


CHAPTER  X. 
THE  DYNAMICS  OF  ELASTIC  BODIES. 

We  shall  now  treat  with  the  help  of  the  principle  of  least  action 
the  rather  complicated  problem  of  the  dynamics  of  continuous  elastic 
media.  Our  considerations  will  extend  the  appreciation  of  the  inti- 
mate relation  between  mass  and  energy  which  we  found  in  our  treat- 
ment of  the  dynamics  of  a  particle.  We  shall  also  be  able  to  show 
that  the  dynamics  of  a  particle  may  be  regarded  as  a  special  case 
of  the  dynamics  of  a  continuous  elastic  medium,  and  to  apply  our 
considerations  to  a  number  of  other  important  problems. 

114.  On  the  Impossibility  of  Absolutely  Rigid  Bodies.     In  the 
older  treatises  on  mechanics,   after  considering  the  dynamics  of  a 
particle  it  was  customary  to  proceed  to  a  discussion  of  the  dynamics 
of  rigid  bodies.     These  rigid  bodies  were  endowed  with  definite  and 
nuchangeable  size  and  shape  and  hence  were  assigned  five  degrees 
of  freedom,  since  it  was  necessary  to  state  the  values  of  five  variables 
completely  to  specify  their  position  in  space.     As  pointed  out  by 
Laue,  however,  our  newer  ideas  as  to  the  velocity  of  light  as  a  limiting 
value  will  no  longer  permit  us  to  conceive  of  a  continuous  body  as 
having  only  a  finite  number  of  degrees  of  freedom.     This  is  evident 
since  it  is  obvious  that  we  could  start  disturbances  simultaneously 
at  an  indefinite  number  of  points  in  a  continuous  body,  and  as  these 
disturbances  cannot  spread  with  infinite  velocity  it  will  be  necessary 
to  give  the  values  of  an  infinite  number  of  variables  in  order  com- 
pletely to  specify  the  succeeding  states  of  the  system.     For  our  newer 
mechanics  the  nearest  approach  to  an  absolutely  rigid  body  would 
of  course  be  one  in  which  disturbances  are  transmitted  with  the 
velocity  of  light.     Since,   then,   the  theory   of  relativity   does   not 
permit  rigid  bodies  we  may  proceed  at  once  to  the  general  theory  of 
deformable  bodies. 

PART  I.     STRESS  AND  STRAIN. 

115.  Definition  of  Strain.     In  the  more  familiar  developments  of 
the  theory  of  elasticity  it  is  customary  to  limit  the  considerations  to 

130 


Dynamics  of  Elastic  Bodies.  131 

the  case  of  strains  small  enough  so  that  higher  powers  of  the  dis- 
placements can  be  neglected,  and  this  introduces  considerable  simpli- 
fication into  a  science  which  under  any  circumstances  is  necessarily 
one  of  great  complication.  Unfortunately  for  our  purposes,  we 
cannot  in  general  introduce  such  a  simplification  if  we  wish  to  apply 
the  theory  of  relativity,  since  in  consequence  of  the  Lorentz  shortening 
a  body  which  appears  unstrained  to  one  observer  may  appear  tre- 
mendously compressed  or  elongated  to  an  observer  moving  with  a 
different  velocity.  The  best  that  we  can  do  will  be  arbitrarily  to 
choose  our  state  of  zero  deformation  such  that  the  strains  will  be 
small  when  measured  in  the  particular  system  of  coordinates  S  in 
which  we  are  specially  interested. 

A  theory  of  strains  of  any  magnitude  was  first  attempted  by 
Saint-  Venant  and  has  been  amplified  and  excellently  presented  by 
Love  in  his  Treatise  on  the  Theory  of  Elasticity,  Appendix  to  Chapter  I. 
In  accordance  with  this  theory,  the  strain  at  any  point  in  a  body  is 
completely  determined  by  six  component  strains  which  can  be  defined 
by  the  following  equations,  wherein  (u,  v,  w)  is  the  displacement  of  a 
point  having  the  unstrained  position  (x,  yy  z)  : 


•99 


£)'  +(5)'). 


_  dw       dv       du  du       dv_  dt>       dw  dw 
€"2  =  <ty  +Hz~*~dy  'dz^"dy~dz^"dy~dzt 

_dw       du       du  du       dv_dv^       dw  dw 
€zi='dx+dz+'dxdz+'dx^~z  +  dx'dz1 

_dv       du       du  du       dv^dv_       dw  dw 
€xv='dx  +  dy  +  'dx'dyJt"dx'dy  +  'dx  dy  ' 

It  will  be  seen  that  these  expressions  for  strain  reduce  to  those 
familiar  in  the  theory  of  small  strains  if  such  second-order  quantities  as. 

(du\2      du  du 
~d~x)  0^  --can  be  neglected. 


132  Chapter  Ten. 

116.  A  physical  significance  for  these  strain  components  will  be 
obtained  if  we  note  that  it  can  be  shown  from  geometrical  considera- 
tions that  lines  which  are  originally  parallel  to  the  axes  have,  when 
strained,  the  elongations 


J*  =   Vl  +  2exx  - 


=  Vl  +  2eyy  -  1,  (149) 


ez 


and  that  the  angles  between  lines  originally  parallel  to  the  axes  are 
given  in  the  strained  condition  by  the  expressions 


cos  6U2  = 


cos  dxz  =  —=^L-=,  (150) 

' 


cos 


Geometrical  considerations  are  also  sufficient  to  show  that  in 
case  the  strain  is  a  simple  elongation  of  amount  e  the  following  equa- 
tion will  be  true  : 


£L  ._       L  -  p    i 

~ 


I2   ~  m2  ~  n2  ~  2mn      2ln      2lm 

where  I,  m,  n  are  the  cosines  which  determine  the  direction  of  the 
elongation. 

117.  Definition  of  Stress.  We  have  just  considered  the  expres- 
sions for  the  strain  at  a  given  point  in  an  elastic  medium;  we  may 
now  define  stress  in  terms  of  the  work  done  in  changing  from  one 
state  of  strain  to  another.  Considering  the  material  contained  in 
-unit  volume  when  the  body  is  unstrained,  we  may  write,  for  the  work 
done  by  this  material  on  its  surroundings  when  a  change  in  strain 
takes  place, 


Dynamics  of  Elastic  Bodies.  133 

dW  =  —  8E  =  tx£5cxx  +  tyyd€vy  +  tzzd€zz 

(152) 

~h  tyzStyz  ~\~  txzfexz  +  txydtXV} 

and  this  equation  serves  to  define  the  stresses  txx,  tvv,  etc.  In  case 
the  strain  varies  from  point  to  point  we  must  consider  of  course  the 
work  done  per  unit  volume  of  the  unstrained  material.  In  case  the 
strains  are  small  it  will  be  noticed  that  the  stresses  thus  defined  are 
identical  with  those  used  in  the  familiar  theories  of  elasticity. 

118.  Transformation  Equations  for  Strain.  We  must  now  prepare 
for  the  introduction  of  the  theory  of  relativity  into  our  considerations, 
by  determining  the  way  the  strain  at  a  given  point  P  appears  to  ob- 
servers moving  with  different  velocities.  Let  the  point  P  in  question 
be  moving  with  the  velocity  u  =  xi  +  yj  +  zk  as  measured  in  sys- 
tem S.  Since  the  state  of  zero  deformation  from  which  to  measure 
strains  can  be  chosen  perfectly  arbitrarily,  let  us  for  convenience 
take  the  strain  as  zero  as  measured  in  system  S,  giving  us 

Cxx  =  €vtf  =  €zz  =  €yz  =  €xz  =  €xy  —  0.  (153) 

What  now  will  be  the  strains  as  measured  by  an  observer  moving 
along  with  the  point  P  in  question?  Let  us  call  the  system  of  coordi- 
nates used  by  this  observer  S°.  It  is  evident  now  from  our  considera- 
tions as  to  the  shape  of  moving  systems  presented  in  Chapter  V  that 
in  system  S°  the  material  in  the  neighborhood  of  the  point  in  question 
will  appear  to  have  been  elongated  in  the  direction  of  motion  in  the 


ratio  of  1  :  -ul  —  —  .     Hence  in  system  S°  the  strain  will  be  an  elonga- 
tion 

e  =  --       =  -  1  •  (154) 


in  the  line  determined  by  the  direction  cosines 

l  =  ^>        m  =  yu>        U  =  Zu-  (155) 

We  may  now  calculate  from  this  elongation  the  components  of 
strain  by  using  equation  (151).     We  obtain 


134 


Chapter  Ten. 


2c2 


eo     _?/£ 
"z  ~  c2 


(156) 


xz 


*»         ,.2 


c2 


and  these  are  the  desired  equations  for  the  strains  at  the  point  P, 
the  accent  °  indicating  that  they  are  measured  with  reference  to  a 
system  of  coordinates  S°  moving  along  with  the  point  itself. 

119.  Variation  in  the  Strain.  We  shall  be  particularly  interested 
in  the  variation  in  the  strain  as  measured  in  S°  when  the  velocity 
experiences  a  small  variation  5u,  the  strains  remaining  zero  as  mea- 
sured in  S.  For  the  sake  of  simplicity  let  us  choose  our  coordinates 
in  such  a  way  that  the  X-axis  is  parallel  to  the  original  velocity,  so 
that  our  change  in  velocity  will  be  from  u  =  xi  to 


=  (x  +  dx)i  +  Byj 


Taking  5u  small  enough  so  that  higher  orders  can  be  neglected,  and 
noting  that  y  =  z  =  0,  we  shall  then  have,  from  equations  (156), 


Dynamics  of  Elastic  Bodies.  135 

1          x 
de°XI  =  -T—      —^r-9  -2  dx,  de°vv  =  0, 


5e°zz  =  0,  dt°v,  =  0,  (157) 

,o  _A      _*„  .o  _J^__*,, 

0€ 


M 


/  W2\C2  "    /  W2\ 

I1-?)  l1--^) 


We  shall  also  be  interested  in  the  variation  in  the  strain  as  measured 
in  S°  produced  by  a  variation  in  the  strain  as  measured  in  S.  Con- 
sidering again  for  simplicity  that  the  X-axis  is  parallel  to  the  motion 
of  the  point,  we  must  calculate  the  variation  produced  in  e°xx,  e°vv, 
etc.,  by  changing  the  values  of  cxx,  cvy,  etc.,  from  zero  to  dexx,  deyv,  etc. 

The  variation  dexx  will  produce  a  variation  in  e°xx  whose  amount 
can  be  calculated  as  follows:  By  equations  (149)  a  line  which  has  unit 
length  and  is  parallel  to  the  X-axis  in  the  unstrained  condition  will 
have  when  strained  the  length  Vl  +  2exx  when  measured  in  system  S 
and  Vl  -f  2€°xx  when  measured  in  system  S°.  Since  the  strain  in 
system  S  is  small,  the  line  remains  sensibly  parallel  to  the  X-axis, 
which  is  also  the  direction  of  motion,  and  these  quantities  will  be 
connected  in  accordance  with  the  Lorentz  shortening  by  the  equation 


=  Jl  -  ^  Vl 

*  C 


(158) 


II  - 
^ 

we  obtain 


Carrying  out  now  our  variation  5€xx,  neglecting  cxx  in  comparison 
with  larger  quantities  and  noting  that  except  for  second  order  quanti- 
ties, 

1    -2c°«  =  ~'=  (159) 


(160) 


Since  the  variations  8evv,  6ezz,  8evz  affect  only  lines  which  are  at 
right  angles  to  the  direction  of  motion,  we  may  evidently  write 

de°vv  =  5evv,          de°z,  =  5ez2,          6e%2  =  devz.  (161) 


136  Chapter  Ten. 

To  calculate  de°xz  we  may  note  that  in  accordance  with  equations 
(150)  we  must  have 


COS  dxz   = 


Vl  +  2exx  Vl  +  2ezz ' 
° 

-O  c     XZ 

cos  6  xz  =     .  = — .  -  , 

Vl  +  2e0,x  Vl  +  2e°,/ 

where  6XZ  is  the  angle  between  lines  which  in  the  unstrained  condition 
are  parallel  to  the  X  and  Z  axes  respectively.  In  accordance  with 
the  Lorentz  shortening,  however,  we  shall  have 


cos  dxz  =  A/I  —  —cos  0  xz' 

Introducing  this  relation,  remembering  that  exx  =  ezz  =  e°22  =  0,  and 
noting  equation  (159),  we  obtain 

fc°M  =  Y~^F\  '  (162) 

and  similarly 


/i«o\ 
A163) 


We  may  now  combine  these  equations  (160),  (161),  (162)  and 
(163)  with  those  for  the  variation  in  strain  with  velocity  and  obtain 
the  final  set  which  we  desire: 

1x1 

5€°-  =  7-   -Ti\i  ;!&*  +  —   TTv56 


de°zz  =  5ezz, 
df°yz  =  5eyz, 

\  x 


('-?)"('-?') 
('-« 


1           X                      1 
«€°«y  =  7 ^¥Y  ^i  8y  +  7 ^iT  8*- 


Dynamics  of  Elastic  Bodies.  137 

These  equations  give  the  variation  in  the  strain  measured  in 
system  S°  at  a  point  P  moving  in  the  X  direction  with  velocity  u, 
provided  the  strains  are  negligibly  small  as  measured  in  S. 

PART  II.     INTRODUCTION  OF  THE  PRINCIPLE  OF  LEAST  ACTION. 

120.  The  Kinetic  Potential  for  an  Elastic  Body.  We  are  now  in 
a  position  to  develop  the  mechanics  of  an  elastic  body  with  the  help 
of  the  principle  of  least  action.  In  Newtonian  mechanics,  as  is  well 
known,  the  kinetic  potential  for  unit  volume  of  material  at  a  given 
point  P  in  an  elastic  body  may  be  put  equal  to  the  density  of  kinetic 
energy  minus  the  density  of  potential  energy,  and  it  is  obvious  that 
our  choice  for  kinetic  potential  must  reduce  to  that  value  at  low 
velocities.  Our  choice  of  an  expression  for  kinetic  potential  is  further- 
more limited  by  the  fundamental  transformation  equation  for  kinetic 
potential  which  we  found  in  the  last  chapter 

(146) 


;  ~v     L    u'2' 
V1  -  7*    V1  -  ^ 


c2 

Taking  these  requirements  into  consideration,  we  may  write  for 
the  kinetic  potential  per  unit  volume  of  the  material  at  a  point  P 
moving  with  the  velocity  u  the  expression 


H  =  -E 


where  E°  is  the  energy  as  measured  in  system  S°  of  the  amount  of 
material  which  in  the  unstrained  condition  (i.  e.,  as  measured  in 
system  S)  is  contained  in  unit  volume. 

The  above  expression  obviously  satisfies  our  fundamental  trans- 
formation equation  (146)  and  at  low  velocities  reduces  in  accordance 
with  the  requirements  of  Newtonian  mechanics  to 

H  =  JmV  -  E°, 

provided  we  introduce  the  substitution  made  familiar  by  our  previous 

E° 

work,  m°  =  —  . 


138 


Chapter  Ten. 


121.  Lagrange's  Equations.  Making  use  of  this  expression  for  the 
kinetic  potential  in  an  elastic  body,  we  may  now  obtain  the  equations 
of  motion  and  stress  for  an  elastic  body  by  substituting  into  Lagrange's 
equations  (139)  Chapter  IX. 

Considering  the  material  at  the  point  P  contained  in  unit  volume 
in  the  unstrained  condition,  we  may  choose  as  our  generalized  co- 
ordinates the  six  component  strains  exx,  eyy,  etc.,  with  the  corre- 
sponding stresses  —  txx,  —  tyy,  etc.,  as  generalized  forces,  and  the 
three  coordinates  x,  y,  z  which  give  the  position  of  the  point  with  the 
corresponding  forces  Fx,  Fy  and  Fz. 

It  is  evident  that  the  kinetic  potential  will  be  independent  of 
the  time  derivatives  of  the  strains,  and  if  we  consider  cases  in  which 
E°  is  independent  of  position,  the  kinetic  potential  will  also  be  inde- 
pendent of  the  absolute  magnitudes  of  the  coordinates  x,  y  and  z. 
Substituting  in  Lagrange's  equations  (139),  we  then  obtain 


-~(  -E°Jl-~]  =  -U 

oCxx  \  C   y 


- 1)-- 


(165) 


(166) 


Dynamics  of  Elastic  Bodies.  139 

We  may  simplify  these  equations,  however;  by  performing  the 
indicated  differentiations  and  making  suitable  substitutions,  we  have 


XX  »*e     XX 


dexx  de  xx  dezx 

But  in  accordance  with  equation  (152)  we  may  write 

8E° 
d€°zx  =        l  xx 

and  from  equations  (164)  we  may  put 


de° 


JCX 


x*  _ 

~  c2 

Making  the  substitutions  in  the  first  of  the  Lagrangian  equations  we 
obtain 


122.  Transformation  Equations  for  Stress.  Similar  substitutions 
can  be  made  in  all  the  equations  of  stress,  and  we  obtain  as  our  set 
of  transformation  equations 


/°  1/2  1/2 

I-  *    .«  .  L  jo 


zz» 


(167) 

*°..  <°x. 


123.  Value  of  E°.  With  the  help  of  these  transformation  equations 
for  stress  we  may  calculate  the  value  of  E°,  the  energy  content,  as 
measured  in  system  S°,  of  material  which  in  the  unstrained  condition 
is  contained  in  unit  volume. 

Consider  unit  volume  of  the  material  in  the  unstrained  condition 
and  call  its  energy  content  w°°.  Give  it  now  the  velocity  u  =  x, 
keeping  its  state  of  strain  unchanged  in  system  S.  Since  the  strain 


140 


Chapter  Ten. 


is  not  changing  in  .system  S,  the  stresses  txx,  etc.,  will  also  be  constant 
in  system  S.  In  system  S°,  however,  the  component  strain  will 
change  in  accordance  with  equations  (156)  from  zero  to 


x* 
2c~2 


and  the  corresponding  stress  will  be  given  at  any  instant  by  the 
expression  just  derived, 


u 


txx  being,  as  we  have  just  seen,  a  constant.     We  may  then  write  for 
E°  the  expression 

"  2c2 


77TO  OO 

Hi     =  w      — 


-~;d 


[('-!) 

Noting  that  u  =  x  we  obtain  on  integration, 

no  oo     i  X1 

E°    =   W°°  +  txx  -  ~ 


(168) 


as  the  desired  expression  for  the  energy  as  measured  in  system  S° 
contained  in  the  material  which  in  system  S  is  unstrained  and  has 
unit  volume. 

124.  The  Equations  of  Motion  in  the  Lagrangian  Form.  We  are 
now  in  a  position  to  simplify  the  three  Lagrangian  equations  (166) 
for  Fx,  Fy  and  Fg.  Carrying  out  the  indicated  differentiation  we  have 


dt 


c2  dx 


and  introducing  the  value  of  E°  given  by  equation  (168)  we  obtain 


txx  x 


(169) 


Dynamics  of  Elastic  Bodies. 


141 


Simple  calculations  will  also  give  us  values  for  Fy  and  Fz.     We  have 
from  (166) 


=  y_         I        rfdl 
,,2  c2      \       c2  a? 


But  since  we  have  adapted  our  considerations  to  cases  in  which  the 
direction  of  motion  is  along  the  X-axis,  we  have  y  =  0;  furthermore 
we  may  substitute,  in  accordance  with  equations  (152),  (157)  and  (167), 


dE°       dE°   de°: 


-  t 


We  thus  obtain  as  our  three  equations  of  motion 

d 


-d-(t     X-} 
~  dt\ txv  &)' 


(170) 


In  these  equations  the  quantities  Fx,  Fv  and  Fz  are  the  components 
of  force  acting  on  a  particular  system,  namely  that  quantity  of  material 
which  at  the  instant  in  question  has  unit  volume.  Since  the  volume 
of  this  material  will  in  general  be  changing,  Fx,  Fv  and  Ft  do  not  give 
us  the  force  per  unit  volume  as  usually  defined.  If  we  represent, 
however,  by  fx,  fv  and  /*  the  components  of  force  per  unit  volume, 
we  may  rewrite  these  equations  in  the  form 

w°°  +  txxx  . 


(171) 


142 


Chapter  Ten. 


where  by  dV  we  mean  a  small  element  of  volume  at  the  point  in 
question. 

125.  Density  of  Momentum.  Since  we  customarily  define  force  as 
equal  to  the  time  rate  of  change  of  momentum,  we  may  now  write  for 
the  density  of  momentum  g  at  a  point  in  an  elastic  body  which  is 
moving  in  the  X  direction  with  the  velocity  u  =  x 


g*   = 


txxx 
c 


v  -j£-j;     w 

^•f  ~z>          &v  =  txy~z 


(172) 


It  is  interesting  to  point  out  that  there  are  components  of  momen- 
tum in  the  Y  and  Z  directions  in  spite  of  the  fact  that  the  material 
at  the  point  in  question  is  moving  in  the  X  direction.  We  shall 
later  see  the  important  significance  of  this  discovery. 

126.  Density  of  Energy.  It  will  be  remembered  that  the  forces 
whose  equations  we  have  just  obtained  are  those  acting  on  unit 
volume  of  the  material  as  measured  in  system  S,  and  hence  we  are 
now  in  a  position  to  calculate  the  energy  density  of  our  material. 
Let  us  start  out  with  unit  volume  of  our  material  at  rest,  with  the 
energy  content  w°°  and  determine  the  work  necessary  to  give  it  the 
velocity  u  =  x  without  change  in  stress  or  strain.  Since  the  only 
component  of  force  which  suffers  displacement  is  FX)  we  have 


w  =  w°°  + 


r.d 

Jo    dt 


X9  x 


xdt, 


]   f   x 

Jo 


t  xd 


1  X 


1       c2 


(173) 


as  an  expression  for  the  energy  density  of  the  elastic  material. 

127.  Summary  of  Results  Obtained  from  the  Principle  of  Least 
Action.  We  may  now  tabulate  for  future  reference  the  results  ob- 
tained from  the  principle  of  least  action. 


Dynamics^  of  Elastic  Bodies.  143 

At  a  given  point  in  an  elastic  medium  which  is  moving  in  the  X 
direction  with  the  velocity  u  =  x,  we  have  for  the  components  of 
stress 


-2     M>        zz  ~   \  ~2     zz> 


(167) 

I    xy 

'"-    r-5 

1    ~~2 


For  the  density  of  energy  at  the  point  in  question  we  have 


W  =  w_JLfr  _  txx.  (173) 


For  the  density  of  momentum  we  have 


x 

2-          (172) 


PART  III.    SOME  MATHEMATICAL  RELATIONS. 

Before  proceeding  to  the  applications  of  these  results  which  we 
have  obtained  from  the  principle  of  least  action,  we  shall  find  it  de- 
sirable to  present  a  number  of  mathematical  relations  which  will 
later  prove  useful. 

128.  The  Unsymmetrical  Stress  Tensor  t.  We  have  defined  the 
components  of  stress  acting  at  a  point  by  equation  (152) 

8W  =  txz8€xx  +  tvv5evv  +  tzz5ezz  +  tVz8eyg  +  txzdexz  +  txyd€xv, 

where  dW  is  the  work  which  accompanies  a  change  in  strain  and  is 
performed  on  the  surroundings  by  the  amount  of  material  which  was 
contained  in  unit  volume  in  the  unstrained  state.  Since  for  con- 
venience we  have  taken  as  our  state  of  zero  strain  the  condition  of 
the  body  as  measured  in  system  S,  it  is  evident  that  the  components 
txx,  tyVJ  etc.,  may  be  taken  as  the  forces  acting  on  the  faces  of  a  unit 
cube  of  material  at  the  point  in  question,  the  first  letter  of  the  sub- 


144  Chapter  Ten. 

script  indicating  the  direction  of  the  force  and  the  second  subscript 
the  direction  of  the  normal  to  the  face  in  question. 

Interpreting  the  components  of  stress  in  this  fashion,  we  may 
now  add  three  further  components  and  obtain  a  complete  tensor 


(174) 


The  three  new  components  tyx,  tzx,  tzy  are  forces  acting  on  the 
unit  cube,  in  the  directions  and  on  the  faces  indicated  by  the  sub- 
scripts. A  knowledge  of  their  value  was  not  necessary  for  our  develop- 
ments of  the  consequences  of  the  principle  of  least  action,  since  it  was 
possible  to  obtain  an  expression  for  the  work  accompanying  a  change 
in  strain  without  their  introduction.  We  shall  find  them  quite  im- 
portant for  our  later  considerations,  however,  and  may  proceed  to 
determine  their  value. 

tyX  is  the  force  acting  in  the  Y  direction  tangentially  to  a  face  of 
the  cube  perpendicular  to  the  X-axis,  and  measured  with  a  system 
of  coordinates  S.  Using  a  system  of  coordinates  S°  which  is  stationary 
with  respect  to  the  point  in  question,  we  should  obtain,  for  the  measure- 
ment of  this  force, 

?,- 

!f! 

in  accordance  with  our  transformation  equation  for  force  (62),  Chapter 
VI.     Similarly  we  shall  have  the  relation 


In  accordance  with  the  elementary  theory  of  elasticity,  however,  the 
forces  t°yx  and  t°xy  which  are  measured  by  an  observer  moving  with 
the  body  will  be  connected  by  the  relation 


xy 


t°xy  being  larger  than  t°yx  in  the  ratio  of  the  areas  of  face  upon  which 
they  act.     Combining  these  three  equations,  and  using  similar  methods 


Dynamics  of  Elastic  Bodies.  145 

for  the  other  quantities,  we  can  obtain  the  desired  relations 


We  see  that  t  is  an  unsymmetrical  tensor. 

129.  The  Symmetrical  Tensor  p.  Besides  this  unsymmetrical  ten- 
sor t  we  shall  find  it  desirable  to  define  a  further  tensor  p  by  the 
equation 

P  =  t  +  gu.  (176) 

We  shall  call  gu  the  tensor  product  of  g  and  u  and  may  indicate 
tensor  products  in  general  by  a  simple  juxtaposition  of  vectors,  gu  is 
itself  a  tensor  with  components  as  indicated  below: 

gxuy    gxuz} 

gvuv    gyuz,  (177) 

gzuy     gzuz. 

Unlike  t.  p  will  be  a  symmetrical  tensor,  since  we  may  show,  by 
substitution  of  the  values  for  g  and  u  already  obtained,  that 

Pyx    =    Pxy,  Pzx    =    PXZ,  P zy    =    Pyz-  (178) 

Consider  for  example  the  value  of  pyx\  we  have  from  our  definition 

Pyx   =    fyz  T  {Jylli, 

and  by  equations  (175)  and  (172)  we  have 

— 

*V  ^2 

and  hence  by  substitution  obtain 

Pyx   =    *zy 

We  also  have,  however,  by  definition 

Pry   =    txy     I     Qx^yj 

and  since  for  the  case  we  are  considering  uv  =  0,  we  arrive  at  the 
equality 

Pxy    =    Pyx. 

The  other  equalities  may  be  shown  in  a  similar  way. 
11 


146  Chapter  Ten. 

130.  Relation  between  div  t  and  tn.  At  a  given  point  P  in  our 
elastic  body  we  shall  define  the  divergence  of  the  tensor  t  by  the  equa- 
tion 

dtxx      dtx       dt 


dt 

(179) 


dt,  dt 


where  i,  j  and  k  are  unit  vectors  parallel  to  the  axes,  div  t  thus  being 
an  ordinary  vector.  It  will  be  seen  that  div  t  is  the  elastic  force 
acting  per  unit  volume  of  material  at  the  point  P. 

Considering  an  element  of  surface  dS,  we  shall  define  a  further 
vector  tn  by  the  equation 

tn  =  (t,x  cos  a  +  txy  cos  ]8  +  txz  cos  7)1 

+   (tyx  COS  a  +  tyy  COS  /3  +  tyz  COS  y)j  (180) 

+  (tzx  cos  a  +  tzv  cos  j8  +  tzz  cos  7)k, 

where  cos  a,  cos  /3  and  cos  7  are  the  direction  cosines  of  the  inward- 
pointing  normal  to  the  element  of  surface  dS. 

Considering  now  a  definite  volume  V  enclosed  by  the  surface  S, 
it  is  evident  that  div  t  and  t»  will  be  connected  by  the  relation 

-  JdivW  =  J%ndS,  (181) 

where  the  symbol  0  indicates  that  the  integration  is  to  be  taken  over 
the  whole  surface  which  encloses  the  volume  V.  This  equation  is 
of  course  merely  a  direct  application  of  Gauss's  formula,  which  states 
in  general  the  equality 


(182) 
(P  cos  a  -f  Q  cos  ]S  +  R  cos  y)dS, 


=    I 

Jo 


where  P,  Q  and  R  may  be  any  functions  of  x,  y  and  z. 


Dynamics  of  Elastic  Bodies.  147 

We  shall  also  find  use  for  a  further  relation  between  div  t  and  tn. 
Consider  a  given  point  of  reference  0,  and  let  r  be  the  radius  vector 
to  any  point  P  in  the  elastic  body;  we  can  then  show  with  the  help 
of  Gauss's  Formula  (182)  that 


jT 


(rXtn)AS 


(txz  -  Uik  +  («*„  - 


where  X  signifies  as  usual   the  outer  product.     Taking   account  of 
equations  (172)  and  (175)  this  can  be  rewritten 

-/(rXdivt)dF=J>Xt,,)dS-/(uXg)dF.          (183) 

131.  The  Equations  of  Motion  in  the  Eulerian  Form.     We  saw  in 

sections  (124)  and  (125)  that  the  equations  of  motion  in  the  Lagran- 
gian  form  might  be  written 

f8V  =  jt 

where  f  is  the  density  of  force  acting  at  any  point  and  g  is  the  density 
of  momentum. 

Provided  that  there  are  no  external  forces  acting  and  f  is  pro- 
duced solely  by  the  elastic  forces,  our  definition  of  the  divergence  of  a 
tensor  will  now  permit  us  to  put 

f  =  -  div  t, 
and  write  for  our  equation  of  motion 

I  (-  div  t>,v  -  £  <««o-.r  *  +  ««£>. 

dg  . 
Expressing  3-  in  terms  of  partial  differentials,  and  putting 


we  obtain 


148  Chapter  Ten. 

Our  symmetrical  tensor  p,  however,  was  defined  by  the  equation  (176) 

p  =  t  +  gu, 

and  hence  we  may  now  write  our  equations  of  motion  in  the  very 
beautiful  Eulerian  form 

-  div  p  =  ^  .  (184) 

We  shall  find  this  simple  form  for  the  equations  of  motion  very 
interesting  in  connection  with  our  considerations  in  the  last  chapter. 

PART  IV.     APPLICATIONS  OF  THE  RESULTS. 

We  may  now  use  the  results  which  we  have  obtained  from  the 
principle  of  least  action  to  elucidate  various  problems  concerning 
the  behavior  of  elastic  bodies. 

132.  Relation  between  Energy  and  Momentum.  In  our  work  on 
the  dynamics  of  a  particle  we  found  that  the  mass  of  a  particle  was 
equal  to  its  energy  divided  by  the  square  of  the  velocity  of  light,  and 
hence  have  come  to  expect  in  general  a  necessary  relation  between 
the  existence  of  momentum  in  any  particular  direction  and  the  trans- 
fer of  energy  in  that  same  direction.  We  find,  however,  in  the  case 
of  elastically  stressed  bodies  a  somewhat  more  complicated  state  of 
affairs  than  in  the  case  of  particles,  since  besides  the  energy  which  is 
transported  bodily  by  the  motion  of  the  medium  an  additional  quan- 
tity of  energy  may  be  transferred  through  the  medium  by  the  action 
of  the  forces  which  hold  it  in  its  state  of  strain.  Thus,  for  example, 
in  the  case  of  a  longitudinally  compressed  rod  moving  parallel  to  its 
length,  the  forces  holding  it  in  its  state  of  longitudinal  compression 
will  be  doing  work  at  the  rear  end  of  the  rod  and  delivering  an  equal 
quantity  of  energy  at  the  front  end,  and  this  additional  transfer  of 
energy  must  be  included  in  the  calculation  of  the  momentum  of  the 
bar. 

As  a  matter  of  fact,  an  examination  of  the  expressions  for  momen- 
tum which  we  obtained  from  the  principle  of  least  action  will  show 
the  justice  of  these  considerations.  For  the  density  of  momentum 
in  the  X  direction  we  obtained  the  expression 

gx  =  (w  +  txx)  - , 

c 


Dynamics  of  Elastic  Bodies.  149 

and  we  see  that  in  order  to  calculate  the  momentum  in  the  X  direc- 
tion we  must  consider  not  merely  the  energy  w  which  is  being  bodily 
carried  along  in  that  direction  with  the  velocity  x,  but  also  must  take 
into  account  the  additional  flow  of  energy  which  arises  from  the 
stress  txx.  As  we  have  already  seen  in  Section  128,  this  stress  txx  can 
be  thought  of  as  resulting  from  forces  which  act  on  the  front  and 
rear  faces  of  a  centimeter  cube  of  our  material.  Since  the  cube  is 
moving  with  the  velocity  x,  the  force  on  the  rear  face  will  do  the 
work  txxx  per  second  and  this  will  be  given  up  at  the  forward  face. 
We  thus  have  an  additional  density  of  energy-flow  in  the  X  direction 
of  the  magnitude  txxx  and  hence  a  corresponding  density  of  momen- 

txxx 
turn  —  . 

Similar  considerations  explain  the  interesting  occurrence  of  com- 
ponents of  momentum  in  the  Y  and  Z  directions, 

x_  x_ 

9*  -  ***  0>         V*  ~  ***  C2» 

in  spite  of  the  fact  that  the  material  involved  is  moving  in  the  X 
direction.  The  stress  txv,  for  example,  can  be  thought  of  as  resulting 
from  forces  which  act  tangentially  in  the  X  direction  on  the  pair  of 
faces  of  our  unit  cube  which  are  perpendicular  to  the  Y  axis.  Since 
the  cube  is  moving  in  the  X  direction  with  the  velocity  x,  we  shall 
have  the  work  txyx  done  at  one  surface  per  second  and  transferred  to 
the  other,  and  the  resulting  flow  of  energy  in  the  X  direction  is  ac- 
companied by  the  corresponding  momentum  -~-  . 

133.  The  Conservation  of  Momentum.  It  is  evident  from  our 
previous  discussions  that  we  may  write  the  equation  of  motion  for 
an  elastic  medium  in  the  form 

«F4*p. 

where  g  is  the  density  of  momentum  at  any  given  point  and  f  is  the 
force  acting  per  unit  volume  of  material.  We  have  already  obtained, 
from  the  principle  of  least  action,  expressions  (172)  which  permit 
the  calculation  of  g  in  terms  of  the  energy  density,  stress  and  velocity 
at  the  point  in  question,  and  our  present  problem  is  to  discuss  some- 
what further  the  nature  of  the  force  f . 


150  Chapter  Ten. 

We  shall  find  it  convenient  to  analyze  the  total  force  per  unit 
volume  of  material  f  into  those  external  forces  fext-  like  gravity,  which 
are  produced  by  agencies  outside  of  the  elastic  body  and  the  internal 
force  tint-  which  arises  from  the  elastic  interaction  of  the  parts  of  the 
strained  body  itself.  It  is  evident  from  the  way  in  which  we  have 
defined  the  divergence  of  a  tensor  (179)  that  for  this  latter  we  may 
write 

tint.  =  -  div  t.  (185) 

Our  equation  of  motion  then  becomes 

(text.  -divt)57  =  —*jp-,  (186) 

or,  integrating  over  the  total  volume  of  the  elastic  body, 

f  (187) 


where  G  is  the  total  momentum  of  the  body.  With  the  help  of  the 
purely  analytical  relation  (181)  we  may  transform  the  above  equation 
into 

f,  (188) 

where  tn  is   defined  in   accordance  with  (180)  so  that  the  integral 
I  tndS  becomes   the  force  exerted  by  the  surroundings  on  the  sur- 

face of  the  elastic  body. 

In  the  case  of  an  isolated  system  both  fext.  and  tn  would  evidently 
be  equal  to  zero  and  we  have  the  principle  of  the  conservation  of 
momentum. 

134.  The  Conservation  of  Angular  Momentum.  Consider  the 
radius  vector  r  from  a  point  of  reference  O  to  any  point  P  in  an  elastic 
body;  then  the  angular  momentum  of  the  body  about  O  will  be 

M  =  /  (r  X  g)dV, 
and  its  rate  of  change  will  be 

?-/('*f)"+/(fx.)".  OS,, 


Dynamics  of  Elastic  Bodies.  151 

Substituting  equation  (186),  this  may  be  written 

-  f  (r  X  f«JdV    -  f  (r  X  div  t)dV  +  J  (u  X  g)dF, 
or,  introducing  the  purely  mathematical  relation  (183)  we  have, 

-  f  (r  X  f^dV  +  J  (r  X  tn)dS.  (190) 

We  see  from  this  equation  that  the  rate  of  change  of  the  angular 
momentum  of  an  elastic  body  is  equal  to  the  moment  of  the  external 
forces  acting  on  the  body  plus  the  moment  of  the  surface  forces. 

In  the  case  of  an  isolated  system  this  reduces  to  the  important 
principle  of  the  conservation  of  angular  momentum. 

135.  Relation  between  Angular  Momentum  and  the  Unsymmetrical 
Stress  Tensor.  The  fact  that  at  a  point  in  a  strained  elastic  medium 
there  may  be  components  of  momentum  at  right  angles  to  the  motion 
of  the  point  itself,  leads  to  the  interesting  conclusion  that  even  in  a 
state  of  steady  motion  the  angular  momentum  of  a  strained  body 
will  in  general  be  changing. 

This  is  evident  from  equation  (189),  in  the  preceding  section, 
which  may  be  written 

(191) 

In  the  older  mechanics  velocity  u  and  momentum  g  were  always  in 
the  same  direction  so  that  the  last  term  of  this  equation  became  zero. 
In  our  newer  mechanics,  however,  we  have  found  (172)  components 
of  momentum  at  right  angles  to  the  velocity  and  hence  even  for  a  body 
moving  in  a  straight  line  with  unchanging  stresses  and  velocity  we  find 
that  the  angular  momentum  is  increasing  at  the  rate 


,  (192) 

and  in  order  to  maintain  the  body  in  its  state  of  uniform  motion  we 
must  apply  external  forces  with  a  turning  moment  of  this  same  amount. 

The  presence  of  this  increasing  angular  momentum  in  a  strained 
body  arises  from  the  unsymmetrical  nature  of  the  stress  tensor,  the  inte- 

gral j  (u  X  g)dV  being  as  a  matter  of  fact  exactly  equal  to  the  integral 


152 


Chapter  Ten. 


over  the  same  volume  of  the  turning  moments  of  the  unsymmetrical 
components  of  the  stress.  Thus,  for  example,  if  we  have  a  body  mov- 
ing in  the  X  direction  with  the  velocity  u  =  xi  we  can  easily  see  from 
equations  (172)  and  (175)  the  truth  of  the  equality 


(u  X  g)  = 


(txz  - 


(txy  -  tyx)ij]. 


B 


136.  The  Right-Angled  Lever.     An    interesting    example    of    the 

principle  that  in  general  a  turning 
1  moment  is  needed  for  the  uniform 
translatory  motion  of  a  strained  body 
is  seen  in  the  apparently  paradoxical 
case  of  the  right-angled  lever. 

Consider  the  right-angled  lever 
shown  in  figure  14.  This  lever  is  sta- 
tionary with  respect  to  a  system  of 
coordinates  S°.  Referred  to  S°  the 
two  lever  arms  are  equal  in  length : 


FIG.  14. 


and  the  lever  is  in  equilibrium  under  the  action  of  the  equal  forces 


Let  us  now  consider  the  equilibrium  as  it  appears,  using  a  system 
of  coordinates  S  with  reference  to  which  the  lever  is  moving  in  X 
direction  with  the  velocity  V.  Referred  to  this  new  system  of  co- 
ordinates the  length  li  of  the  arm  which  lies  in  the  Y  direction  will  be 
the  same  as  in  system  >S°,  giving  us 


But  for  the  other  arm  which  lies  in  the  direction  of  motion  we  shall 
have,  in  accordance  with  the  Lorentz  shortening, 


2 


For  the  forces  FI  and  Fz  we  shall  have,  in  accordance  with  our  equa- 


Dynamics  of  Elastic  Bodies.  153 

tions  for  the  transformation  of  force  (61)  and  (62), 


F2  =  F 
We  thus  obtain  for  the  moment  of  the  forces  around  the  pivot  B 

rr    7  JJT    7  77»    07    O  T7T    OT    O    I      -I  I     -        J?    °7    °  _     77    7 

r  ill  —  r  zlz  —  r  i  li    •-  r  2  &2    I  1  -  -  ~~^~  I  —  i*  i  &i      2   —  r  i^i  —  , 

and  are  led  to  the  remarkable  conclusion  that  such  a  moving  lever 
will  be  in  equilibrium  only  if  the  external  forces  have  a  definite  turning 
moment  of  the  magnitude  given  above. 

The  explanation  of  this  apparent  paradox  is  obvious,  however, 
in  the  light  of  our  previous  discussion.  In  spite  of  the  fact  that  the 
lever  is  in  uniform  motion  in  a  straight  line,  its  angular  momentum 
is  continually  increasing  owing  to  the  fact  that  it  is  elastically  strained, 
and  it  can  be  shown  by  carrying  out  the  integration  indicated  in 

equation  (192)  that  the  rate  of  change  of  angular  momentum  is  as  a 

72 

matter  of  fact  just  equal  to  the  turning  moment  Fill  — . 

c 

72 
This  necessity  for  a  turning  moment  Fill  —  can  also  be  shown 

directly  from  a  consideration  of  the  energy  flow  in  the  lever.  Since 
the  force  FI  is  doing  the  work  F\V  per  second  at  the  point  A,  a  stream 
of  energy  of  this  amount  is  continually  flowing  through  the  lever 
from  A  to  the  pivot  B.  In  accordance  with  our  ideas  as  to  the  rela- 
tion between  energy  and  mass,  this  new  energy  which  enters  at  A  each 

FiV 

second  has  the  mass  — £- ,  and  hence  each  second  the  angular  mo- 
c 

mentum  of  the  system  around  the  point  B  is  increased  by  the  amount 


We  have  already  found,  however,  exactly  this  same  expression  for 
the  moment  of  the  forces  around  the  pivot  B  and  hence  see  that  they 
are  of  just  the  magnitude  necessary  to  keep  the  lever  from  turning, 
thus  solving  completely  our  apparent  paradox. 


154 


Chapter  Ten. 


137.  Isolated  Systems  in  a  Steady  State.  Our  considerations  have 
shown  that  the  density  of  momentum  is  equal  to  the  density  of  energy 
flow  divided  by  the  square  of  the  velocity  of  light.  If  we  have  a 
system  which  is  in  a  steady  internal  state,  and  is  either  isolated  or 
merely  subjected  to  an  external  pressure  with  no  components  of  force 
tangential  to  the  bounding  surface,  it  is  evident  that  the  resultant 
flow  of  energy  for  the  whole  body  must  be  in  the  direction  of  motion, 
and  hence  for  these  systems  momentum  and  velocity  will  be  in  the 
same  direction  without  the  complications  •  introduced  by  a  trans- 
verse energy  flow. 

Thus  for  an  isolated  system  in  a  steady  internal  state  we  may 
write, 

R* 

V  ~2 

(193) 


138.  The  Dynamics  of  a  Particle.  It  is  important  to  note  that 
particles  are  interesting  examples  of  systems  in  which  there  will 
obviously  be  no  transverse  component  of  energy  flow  since  their 
infinitesimal  size  precludes  the  action  of  tangential  surface  forces. 
We  thus  see  that  the  dynamics  of  a  particle  may  be  regarded  as  a 
special  case  of  the  more  general  dynamics  which  we  have  developed 
in  this  chapter,  the  equation  of  motion  for  a  particle  being 


m 


dt 


in  agreement  with  the  work  of  Chapter  VI. 

139.  Conclusion.  We  may  now  point  out  in  conclusion  the  chief 
results  of  this  chapter.  With  the  help  of  Einstein's  equations  for 
spatial  and  temporal  considerations,  we  have  developed  a  set  of 
transformation  equations  for  the  strain  in  an  elastic  body.  Using  the 
components  of  strain  and  velocity  as  generalized  coordinates,  we  then 
introduced  the  principle  of  least  action,  choosing  a  form  of  function 


Dynamics  of  Elastic  Bodies.  155 

for  kinetic  potential  which  agrees  at  low  velocities  with  the  choice 
made  in  the  older  theories  of  elasticity  and  at  all  velocities  agrees 
with  the  requirements  of  the  principle  of  relativity.  Using  the 
Lagrangian  equations,  we  were  then  able  to  develop  all  that  is  neces- 
sary for  a  complete  theory  of  elasticity. 

The  most  important  consequence  of  these  considerations  is  an 
extension  in  our  ideas  as  to  the  relation  between  momentum  and 
energy.  We  find  that  the  density  of  momentum  in  any  direction 
must  be  placed  equal  to  the  total  density  of  energy  flow  in  that  same 
direction  divided  by  the  square  of  the  velocity  of  light;  and  we  find 
that  we  must  include  in  our  density  of  energy  flow  that  transferred 
through  the  elastic  body  by  the  forces  which  hold  it  in  its  state  of 
strain  and  suffer  displacement  as  the  body  moves.  This  involves  in 
general  a  flow  of  energy  and  hence  momentum  at  right  angles  to  the 
motion  of  the  body  itself. 

At  present  we  have  no  experiments  of  sufficient  accuracy  so  that 
we  can  investigate  the  differences  between  this  new  theory  of  elasticity 
and  the  older  ones,  and  hence  of  course  have  found  no  experimental 
contradiction  to  the  new  theory.  It  will  be  seen,  however,  from  the 
expressions  for  momentum  that  even  at  low  velocities  the  conse- 
quences of  this  new  theory  will  become  important  as  soon  as  we 
run  across  elastic  systems  in  which  very  large  stresses  are  involved. 
It  is  also  important  to  show  that  a  theory  of  elasticity  can  be  de- 
veloped which  agrees  with  the  requirements  of  the  theory  of  relativity. 
In  fairness,  it  must,  however,  be  pointed  out  in  conclusion  that  since 
our  expression  for  kinetic  potential  was  not  absolutely  uniquely  deter- 
mined there  may  also  be  other  theories  of  elasticity  which  will  agree 
with  the  principle  of  relativity  and  with  all  the  facts  as  now  known. 


CHAPTER  XI. 
THE  DYNAMICS  OF  A  THERMODYNAMIC  SYSTEM. 

We  may  now  use  our  conclusions  as  to  the  relation  between  the 
principle  of  least  action  and  the  theory  of  relativity  to  obtain  informa- 
tion as  to  the  behavior  of  thermodynamic  systems  in  motion. 

140.  The  Generalized  Coordinates  and  Forces.     Let  us  consider  a 
thermodynamic  system   whose   state   is   defined   by   the   generalized 
coordinates  volume  v,  entropy  S  and  the  values  of  x,  y  and  z  which 
determine  its  position.     Corresponding  to  these  coordinates  we  shall 
have  the  generalized  external  forces,  the  negative  of  the  pressure, 

-  p,  temperature,  T,  and  the  components  of  force,  FX)  Fy  and  F  z. 
These  generalized  coordinates  and  forces  are  related  to  the  energy 
change  dE  accompanying  a  small  displacement  d,  in  accordance  with 
the  equation 

dE  =  -  dW  =  -  pdv  +  T5S  +  Fxdx  +  Fydy  +  Fz5z.     (194) 

141.  Transformation  Equation  for  Volume.     Before  we  can  apply 
the  principle  of  least  action  we  shall  need  to  have  transformation 
equations  for  the  generalized  coordinates,  volume  and  entropy. 

In  accordance  with  the  Lorentz  shortening,  we  may  write  the 
following  expression  for  the  volume  v  of  the  system  in  terms  of  v°  as 
measured  with  a  set  of  'axes  S°  with  respect  to  which  the  system  is 
stationary  : 


where  u  is  the  velocity  of  the  system. 

By  differentiation  we  may  obtain  expressions  which  we  shall  find 
useful, 


dx 

156 


Dynamics  of  a  Thermodynamic  System.  157 

142.  Transformation  Equation  for  Entropy.     As  for  the  entropy 
of  a  thermodynamic  system,  this  is  a  quantity  which  must  appear 
the  same  to  all  observers  regardless  of  their  motion.     This  invariance 
of  entropy  is  a  direct  consequence  of  the  close  relation  between  the 
entropy  of  a  system  in  a  given  state  and  the  probability  of  that  state. 
Let  us  write,  in  accordance  with  the  Boltzmann-Planck  ideas  as  to 
the  interdependence  of  these  quantities, 

S  =  k  log  TF, 

where  S  is  the  entropy  of  the  system  in  the  state  in  question,  k  is  a 
universal  constant,  and  W  the  probability  of  having  a  microscopic 
arrangement  of  molecules  or  other  elementary  constituent  parts  which 
corresponds  to  the  desired  thermodynamic  state.  Since  this  prob- 
ability is  evidently  independent  of  the  relative  motion  of  the  observer 
and  the  system  we  see  that  the  entropy  of  a  system  S  must  be  an 
invariant  and  may  write 

S  =  S°.  (197) 

143.  Introduction  of  the  Principle  of  Least  Action.    The  Kinetic 
Potential.     We  are  now  in  a  position  to  introduce  the  principle  of 
least  action  into  our  considerations  by  choosing  a  form  of  function 
for  the  kinetic  potential  which  will  agree  at  low  velocities  with  the 
familiar  principles  of  thermodynamics  and  will  agree  at  all  velocities 
with  the  requirements  of  the  theory  of  relativity.* 

If  we  use  volume  and  entropy  as  our  generalized  coordinates,  these 
conditions  are  met  by  taking  for  kinetic  potential  the  expression 


(198) 


This  expression  agrees  with  the  requirements  of  the  theory  of 

TT 

relativity  that  — .  shall  be  an  invariant  (see  Section  111)  and 


at  low  velocities  reduces  to  H  =  —  E,  which  with  our  choice  of 
coordinates  is  the  familiar  form  for  the  kinetic  potential  of  a  thermo- 
dynamic system. 


158  Chapter  Eleven. 

It  should  be  noted  that  this  expression  for  the  kinetic  potential 
of  a  thermodynamic  system  applies  of  course  only  provided  we  pick 
out  volume  v  and  entropy  S  as  generalized  coordinates.  If,  following 
Helmholtz,  we  should  think  it  more  rational  to  take  v  as  one  coordinate 
and  a  quantity  6  whose  time  derivative  is  equal  to  temperature, 
0  =  T7,  as  the  other  coordinate,  we  should  obtain  of  course  a  different 
expression  for  the  kinetic  potential;  in  fact  should  have  under  those 
circumstances 


H  =  (E°  -  T°S°)      1  ~  7  ' 

Using  this  value  of  kinetic  potential,  however,  with  the  corresponding 
coordinates  we  should  obtain  results  exactly  the  same  as  those  which 
we  are  now  going  to  work  out  with  the  help  of  the  other  set  of  coordi- 
nates. 

144.  The  Lagrangian  Equations.  Having  chosen  a  form  for  the 
kinetic  potential  we  may  now  substitute  into  the  Lagrangian  equa- 
tions (139)  and  obtain  the  desired  information  with  regard  to  the 
behavior  of  thermodynamic  systems. 

Since  we  shall  consider  cases  in  which  the  energy  of  the  system  is 
independent  of  the  position  in  space,  the  kinetic  potential  will  be 
independent  of  the  coordinates  x,  y  and  2,  depending  only  on  their 
time  derivatives.  Noting  also  that  the  kinetic  potential  is  inde- 
pendent of  the  time  derivatives  of  volume  and  entropy,  we  shall 
obtain  the  Lagrangian  equations  in  the  simple  form 


(199) 


Dynamics  of  a  Thermodynamic  System.  159 

145.  Transformation  Equation  for  Pressure.  We  may  use  the  first 
of  these  equations  to  show  that  the  pressure  is  a  quantity  which 
appears  the  same  to  all  observers  regardless  of  their  relative  motion. 
We  have 

~tt*  dE°  dvc 
dv°  dv  ' 

dE° 

But,  in  accordance  with   equation    (194),  p°  =  —  -r~^ .  and  in  ac- 

ov    ' 

cordance  with  equation  (195), 

dv°  1 


dv 


r  *• 

V1-  * 


which  gives  us  the  desired  relation 

p  =  p°.  (200) 

Defining  pressure  as  force  per  unit  area,  this  result  will  be  seen 
to  be  identical  with  that  which  is  obtained  from  the  transformation 
equations  for  force  and  area  which  result  from  our  earliest  considera- 
tions. 

146.  Transformation  Equation  for  Temperature.  The  second  of 
the  Lagrangian  equations  (199)  will  provide  us  information  as  to 
measurements  of  temperature  made  by  observers  moving  with  different 
velocities.  We  have 


r 

But,  in  accordance  with  equation  (194),  —  ^  =  T°  and  in  accordance 

dS° 
with  (197)  -r-~-  =  1.     We  obtain  as  our  transformation  equation, 


T  =  T°  \\i-~,  (201) 

T 

and  see  that  the  quantity      .  is  an  invariant  for  the  Lorentz 


transformation 


160  Chapter  Eleven. 

147.  The  Equations  of  Motion  for  Quasistationary  Adiabatic 
Acceleration.  Let  us  now  turn  our  attention  to  the  last  three  of  the 
Lagrangian  equations.  These  are  the  equations  for  the  motion  of  a 
thermodynamic  system  under  the  action  of  external  force.  It  is 
evident,  however,  that  these  equations  will  necessarily  apply  only 
to  cases  of  quasistationary  acceleration,  since  our  development  of 
the  principle  of  least  action  gave  us  an  equation  for  kinetic  potential 
which  was  true  only  for  systems  of  infinitesimal  extent  or  large  systems 
in  a  steady  internal  state.  It  is  also  evident  that  we  must  confine  our 
considerations  to  cases  of  adiabatic  acceleration,  since  otherwise  the 
value  of  E°  which  occurs  in  the  expression  for  kinetic  potential  might 
be  varying  in  a  perfectly  unknown  manner. 

The  Lagrangian  equations  for  force  may  be  advantageously  trans- 
formed. We  have 


But  by  equations  (194),  (196)  and  (197)  we  have 

dE°  dv°  v°         x  dS° 

ST-.-tfi        Zr-T-*     -2-YT2>        and        —7  =  0. 


We  obtain 


(202) 


Similar  equations  may  be  obtained  for  the  components  of  force  in 
the  Y  and  Z  directions  and  these  combined  to  give  the  vector  equation 

d  fg°  +  pVu] 
~  dt\      I        «2c4'  (203) 

~ 


Dynamics  of  cr  Thermodynamic  System. 


161 


This  is  the  fundamental  equation  of  motion  for  the  dynamics  of  a 
thermodynamic  system. 

148.  The  Energy  of  a  Moving  Thermodynamic  System.  We  may 
use  this  equation  to  obtain  an  expression  for  the  energy  of  a  moving 
thermodynamic  system.  If  we  adiabatically  accelerate  a  thermo- 
dynamic system  in  the  direction  of  its  motion,  its  energy  will  increase 
both  because  of  the  work  done  by  the  force 


__ 

= 


which  produces  the  acceleration  and  because  of  the  work  done  by  the 
pressure  p  =  p°  which  acts  on  a  volume'  which  is  continually  dimin- 
ishing as  the  velocity  u  increases,  in  accordance  with  the  expression 

/         u2 
v  =  v°  A/I  -   —  .     Hence  we  may  write  for  the  total  energy 


r-* 

V1'  7* 


(204) 


149.  The  Momentum  of  a  Moving  Thermodynamic  System.  We 
may  compare  this  expression  for  the  energy  of  a  thermodynamic 
system  with  the  following  expression  for  momentum  which  is  evident 
from  the  equation  (203)  for  force: 


(205) 


We  find  again,  as  in  our  treatment  of  elastic  bodies  presented 
in  the  last  chapter,  that  the  momentum  of  a  moving  system  may  be 
calculated  by  taking  the  total  flow  of  energy  in  the  desired  direction 

12 


162 


Chapter  Eleven. 


and  dividing  by  c2.     Thus,   comparing  equations   (204)   and   (205), 
we  have 


„       i      £_ j.| 

~  c2      "r  c2     ' 


(206) 


where  the  term  -5  u  takes  care  of  the  energy  transported  bodily  along 
c 

by  the  system  and  the  term  —  u  takes  care  of  the  energy  transferred 

c 

in  the  u  direction  by  the  action  of  the  external  pressure  on  the  rear 
and  front  end  of  the  moving  system. 

150.  The  Dynamics  of  a  Hohlraum.  As  an  application  of  our  con- 
siderations we  may  consider  the  dynamics  of  a  hohlraum,  since  a 
hohlraum  in  thermodynamic  equilibrium  is  of  course  merely  a  special 
example  of  the  general  dynamics  which  we  have  just  developed.  The 
simplicity  of  the  hohlraum  and  its  importance  from  a  theoretical 
point  of  view  make  it  interesting  to  obtain  by  the  present  method  the 
same  expression  for  momentum  that  can  be  obtained  directly  but 
with  less  ease  of  calculation  from  electromagnetic  considerations. 

As  is  well  known  from  the  work  of  Stefan  and  Boltzmann,  the 
energy  content  E°  and  pressure  p°  of  a  hohlraum  at  rest  and  in  thermo- 
dynamic equilibrium  are  completely  determined  by  the  temperature 
T°  and  volume  v°  in  accordance  with  the  equations 


where  a  is  the  so-called  Stefan's  constant. 

Substituting  these  values  of  E°  and  p°  in  the  equation  for  the 
motion  of  a  thermodynamic  system  (203),  we  obtain 


d_ 
dt 


avT* 


(207) 


as  the  equation  for  the  quasistationary  adiabatic  acceleration  of  a 


Dynamics  of  a  Thermodynamic  System.  163 

hohlraum.     In  view  of  this  equation  we  may  write  for  the  momentum 
of  a  hohlraum  the  expression 

4    av°T°*    u 
G-3—  =.*'  (208) 


It  is  a  fact  of  significance  that  our  dynamics  leads  to  a  result  for 
the  momentum  of  a  hohlraum  which  had  been  adopted  on  the  ground 
of  electromagnetic  considerations  even  without  the  express  intro- 
duction of  relativity  theory. 


CHAPTER   XII. 
ELECTROMAGNETIC  THEORY. 

The  Einstein  theory  of  relativity  proves  to  be  of  the  greatest 
significance  for  electromagnetics.  On  the  one  hand,  the  new  electro- 
magnetic theory  based  on  the  first  postulate  of  relativity  obviously 
accounts  in  a  direct  and  straightforward  manner  for  the  results  of  the 
Michelson-Morley  experiment  and  other  unsuccessful  attempts  to 
detect  an  ether  drift,  and  on  the  other  hand  also  accounts  just  as 
simply  for  the  phenomena  of  moving  dielectrics  as  did  the  older 
theory  of  a  stationary  ether.  Furthermore,  the  theory  of  relativity 
provides  considerably  simplified  methods  for  deriving  a  great  many 
theorems  which  were  already  known  on  the  basis  of  the  ether  theory, 
and  gives  us  in  general  a  clarified  insight  into  the  nature  of  electro- 
magnetic action. 

151.  The  Form  of  the  Kinetic  Potential.  In  Chapter  IX  we  in- 
vestigated the  general  relation  between  the  principle  of  least  action 
and  the  theory  of  the  relativity  of  motion.  We  saw  that  the  develop- 
ment of  any  branch  of  dynamics  would  agree  with  the  requirements 
of  relativity  provided  only  that  the  kinetic  potential  H  has  such  a  form 

TT 

that  the  quantity      ,  is  an  invariant  for  the  Lorentz  transfor- 


mation.  Making  use  of  this  discovery  we  have  seen  the  possibility 
of  developing  the  dynamics  of  a  particle,  the  dynamics  of  an  elastic 
body,  and  the  dynamics  of  a  thermodynamic  system,  all  of  them  in 
forms  which  agree  with  the  theory  of  relativity  by  merely  introducing 
slight  modifications  into  the  older  expressions  for  kinetic  potential  in 

TT 

such  a  way  as  to  obtain  the  necessary  invariance  for 


In  the  case  of  electrodynamics,  however,  on  account  of  the  closely 
interwoven  historical  development  of  the  theories  of  electricity  and 
relativity,  we  shall  not  find  it  necessary  to  introduce  any  modification 

164 


Electromagnetic  Theory.  165 

in  the  form  of  the  kinetic  potential,  but  may  take  for  H  the  following 
expression,  which  is  known  to  lead  to  the  familiar  equations  of  the 
Lorentz  electron  theory 

(209) 

where  the  integration  is  to  extend  over  the  whole  volume  of  the 
system  V,  e  is  the  intensity  of  the  electric  field  at  the  point  in  question, 
<t>  is  the  value  of  the  vector  potential,  p  the  density  of  charge  and  u  its 
velocity.* 

Let  us  now  show  that  the  expression  which  we  have  chosen  for 
kinetic  potential  does  lead  to  the  familiar  equations  of  the  electron 
theory. 

152.  The  Principle  of  Least  Action.  If  now  we  denote  by  f  the 
force  per  unit  volume  of  material  exerted  by  the  electromagnetic 
action  it  is  evident  that  we  may  write  in  accordance  with  the  principle 
of  least  action  (135) 

),      (210) 


where  6r  is  the  variation  in  the  radius  vector  to  the  particle  under 
consideration,  and  where  the  integration  is  to  be  taken  over  the 
whole  volume  occupied  by  the  system  and  between  two  instants  of 
time  ti  and  tz  at  which  the  actual  and  displaced  configurations  of  the 
system  coincide. 

153.  The  Partial  Integrations.  In  order  to  simplify  this  equation, 
we  shall  need  to  make  use  of  two  results  which  can  be  obtained  by 
partial  integrations  with  respect  to  time  and  space  respectively. 

Thus  we  may  write 


*  Strictly  speaking  this  expression  for  kinetic  potential  is  not  quite  correct, 
since  kinetic  potential  must  have  the  dimensions  of  energy.  To  complete  the  equa- 
tion and  give  all  the  terms  their  correct  dimensions,  we  could  multiply  the  first  term 
by  the  dielectric  inductivity  of  free  space  e,  and  the  last  two  terms  by  the  magnetic 
permeability  M-  Since,  however,  c  and  n  have  the  numerical  value  unity  with  the 
usual  choice  of  units,  we  shall  not  be  led  into  error  in  our  particular  considerations 
if  we  omit  these  factors. 


166  Chapter  Twelve. 

or,  since  the  displaced  and  actual  motions  coincide  at  ti  and  £2, 

f<fe(aa&)  =  --J  '  dt(~db}  (211) 


We  may  also  write 

dv    a        = 


or,  since  we  are  to  carry  out  our  integrations  over  the  whole  volume 
occupied  by  the  system,  we  shall  take  our  functions  as  zero  at  the 
limits  of  integration  and  may  write 

(212) 

Since  similar  considerations  apply  to  derivatives  with  respect  to  the 
other  variables  y  and  z,  we  can  also  obtain 

fdVadivb  =  -/dFb-grada,  (213) 

fdVsi-  curl  b  =  /  dV  b  -  curl  a.  (214) 

154.  Derivation  of  the  Fundamental  Equations  of  Electromagnetic 
Theory.  Making  use  of  these  purely  mathematical  relationships  we 
are  now  in  a  position  to  develop  our  fundamental  equation  (210). 

Carrying  out  the  indicated  variation,  noting  that  5u  =  ,  and 
making  use  of  (211)  and  (214)  we  easily  obtain 

i  £}•«.  +  {curl  curl*-  (5+  P2)}.  * 

--S(pu)  +f-Sr     =  0. 

In  developing  the  consequences  of  this  equation,  it  should  be 
noted,  however,  that  the  variations  are  not  all  of  them  independent; 
thus,  since  we  shall  define  the  density  of  charge  by  the  equation 

P  =  div  e,  (216) 

it  is  evident  that  it  will  be  necessary  to  preserve  the  truth  of  this 
equation  in  any  variation  that  we  carry  out.  This  can  evidently  be 


Electromagnetic  Theory.  167 

done  if  we  add  to  our  equation  (215)  the  expression 
/  dtdV\l/[dp  -  div  6e]  =  0, 

where  ^  is  an  undetermined  scalar  multiplier.     We  then  obtain  with 
the  help  of  (213) 

f  I          !  *+  i  ,1  L  I 

\  e  -+-  — —  +  grad  ^  f  •  de  +  i  curl  curl  9 

I    I  C    ot  J  L 

(217) 
=  0, 


and  may  now  treat  the  variations  de  and  6<(>  as  entirely  independent 
of  the  others;  we  must  then  have  the  following  equations  true 

e=  -^-grad^,  (218) 

curl  curl  +  =  -  +  ^ ,  (219) 

and  have  thus  derived  from  the  principle  of  least  action  the  funda- 
mental equations  of  modern  electron  theory.  We  may  put  these  in 
their  familiar  form  by  defining  the  magnetic  field  strength  h  by  the 
equation 

h  =  curl  4  (220) 

We  then  obtain  from  (219) 

curlh  =  if+pH,  (221) 

and,  noting  the  mathematical  identity  curl  grad  \J/  =  0,  we  obtain 
from  (218) 

curie  --if.  (222) 

We  have  furthermore  by  definition  (216) 

div  e  =  P,  (223) 

and  noting  equation  (220)  may  write  the  mathematical  identity 

div  h  =  0.  (224) 


168  Chapter  Twelve. 

These  four  equations  (221-4)  are  the  familiar  expressions  which 
have  been  made  the  foundation  of  modern  electron  theory.  They 
differ  from  Maxwell's  original  four  field  equations  only  by  the  intro- 
duction in  (221)  and  (223)  of  terms  which  arise  from  the  density  of 
charge  p  of  the  electrons,  and  reduce  to  Maxwell's  set  in  free  space. 

155.  We  have  not  yet  made  use  of  the  last  three  terms  in  the 
fundamental  equation  (217)  which  results  from  the  principle  of  least 
action.  As  a  matter  of  fact,  it  can  be  shown  that  these  terms  can  be 
transformed  into  the  expression 


fdtdV  [f  ^  -  f  [uX  curl  4>]*  +  P  grad  ^  +  f  1  -5r, 


(225) 


and  hence  lead  to  the  familiar  fifth  fundamental  equation  of  modern 
electron  theory, 

f      6<t>  [~u  L1*l 

f  =  p  |  -  —  -  grad  ^  +     -  X  curl  9       J  > 

f  =  p{e+  [*Xh]*}.  (226) 

The  transformation  of  the  last  three  terms  of  (217)  into  the  form 
given  above  (225)  is  a  complicated  one  and  it  has  not  seemed  neces- 
sary to  present  it  here  since  in  a  later  paragraph  we  shall  show  the 
possibility  of  deriving  the  fifth  fundamental  equation  of  the  electron 
theory  (226)  by  combining  the  four  field  equations  (221-4)  with  the 
transformation  equations  for  force  already  obtained  from  the  principle 
of  relativity.  The  reader  may  carry  out  the  transformation  himself, 
however,  if  he  makes  use  of  the  partial  integrations  which  we  have 
already  obtained,  notes  that  in  accordance  with  the  principle  of  the 
conservation  of  electricity  we  must  have  dp  =  —  divp  5r  and  notes 

that  5u  =  —  7—  ,  where  the  differentiation  --  indicates  that  we  are 
dt  dt 

following  some  particular  particle  in  its  motion,  while  the  differentia- 

A  /}ft» 

tion  --  occurring  in  -  -  indicates  that  we  intend  the  rate  of  change 
dt  ot 

at  some  particular  stationary  point. 

156.  The  Transformation  Equations  for  e,  h  and  p.     We  have  thus 
shown  the  possibility  of  deriving  the  fundamental  equations  of  modern 


Electromagnetic  Theory.  169 

electron  theory  from  the  principle  of  least  action.  We  now  wish  to 
introduce  the  theory  of  relativity  into  our  discussions  by  presenting 
a  set  of  equations  for  transforming  measurements  of  e,  h  and  p  from 
one  set  of  space-time  coordinates  S  to  another  set  S'  moving  past  S 
in  the  X-direction  with  the  velocity  V.  This  set  of  equations  is  as 
follows  : 

e*   =  ex, 


(227) 

e,'  =  *(  ez  +  —h*J> 

h,'  =  hx, 

hv'  =  '*»+*,  (228) 


h,'  =  K  f  h,  -  —  ey  }  , 

(229) 


where  K  has  its  customary  significance 


As  a  matter  of  fact,  this  set  of  transformation  equations  fulfills 
all  the  requirements  imposed  by  the  theory  of  relativity.  Thus,  in 
the  first  place,  it  will  be  seen,  on  development,  that  these  equations 
are  themselves  perfectly  symmetrical  with  respect  to  the  primed  and 
unprimed  quantities  except  for  the  necessary  change  from  +  V  to 
—  V.  In  the  second  place,  it  will  be  found  that  the  substitution  of 
these  equations  into  our  five  fundamental  equations  for  electro- 
magnetic theory  (221-2-3-4-6)  will  successfully  transform  them 
into  an  entirely  similar  set  with  primed  quantities  replacing  the 
unprimed  ones.  And  finally  it  can  be  shown  that  these  equations 
agree  with  the  general  requirement  derived  in  Chapter  IX  that  the 


170  Chapter  Twelve. 

TT 

quantity       ,  shall   be   an  invariant    for    the    Lorentz    trans- 


formation. 

TT 

To  demonstrate  this  important  invariance  of  —  .  we  may 

u2 


point  out  that  by  introducing  equations  (220),  (221)  and  (214),  our 
original  expression  for  kinetic  potential 

i  (curl<|>) 

- 

can  easily  be  shown  equal  to 


(230) 

and,  noting  that  our  fundamental  equations  for  space  and  time  pro- 
vide us  with  the  relation 

dV  dV 


we  can  easily  show  that  our  transformation  equations  for  e  and  h  do 

lead  to  the  equality 

H  H' 


We  thus  know  that  our  development  of  the  fundamental  equations 
for  electromagnetic  theory  from  the  principle  of  least  action  is  indeed 
in  complete  accordance  with  the  theory  of  relativity,  since  it  conforms 
with  the  general  requirement  which  was  found  in  Chapter  IX  to  be 
imposed  by  the  theory  of  relativity  on  all  dynamical  considerations. 

157.  The  Invariance  of  Electric  Charge.  As  to  the  significance  of 
the  transformation  equations  which  we  have  presented  for  e,  h  and  p, 
we  may  first  show,  in  accordance  with  the  last  of  these  equations, 
that  a  given  electric  charge  will  appear  the  same  to  all  observers  no 
matter  what  their  relative  motion. 


Electromagnetic  Theory.  171 

To  demonstrate  this  we  merely  have  to  point  out  that,  by  intro- 
ducing equation  (17),  we  may  write  our  transformation  equation 
for  p  (229)  in  the  form 


Pf 


which  shows  at  once  that  the  two  measurements  of  density  of  charge 
made  by  0  and  0'  are  in  exactly  the  same  ratio  as  the  corresponding 
measurements  for  the  Lorentz  shortening  of  the  charged  body,  so 
that  the  total  charge  will  evidently  measure  the  same  for  the  two 
observers. 

We  might  express  this  invariance  of  electric  charge  by  writing  the 
equation 

Q'  =  Q.  (231) 

It  should  be  noted  in  passing  that  this  result  is  in  entire  accord 
with  the  whole  modern  development  of  electrical  theory,  which  lays 
increasing  stress  on  the  fundamentally  and  indivisibility  of  the 
electron  as  the  natural  unit  quantity  of  electricity.  On  this  basis 
the  most  direct  method  of  determining  the  charge  on  an  electrified 
body  would  be  to  count  the  number  of  electrons  present  and  this 
number  must  obviously  appear  the  same  both  to  observer  0  and 
observer  0'.* 

158.  The  Relativity  of  Magnetic  and  Electric  Fields.  As  to  the 
significance  of  equations  (227)  and  (228)  for  transforming  the  values 
of  the  electric  and  magnetic  field  strengths  from  one  system  to  another, 
we  see  that  at  a  given  point  in  space  we  may  distinguish  between  the 
electric  vector  e  =  exi  +  e  J  -f  ezk  as  measured  by  our  original 
observer  0  and  the  vector  e'  =  ex'i  +  ev'j  +  ez'k  as  measured  in 
units  of  his  own  system  by  an  observer  0'  who  is  moving  past  0  with 
the  velocity  V  in  the  X-direction.  Thus  if  0  finds  in  an  unvarying 
electromagnetic  field  that  Qe  is  the  force  on  a  small  test  charge  Q 
which  is  stationary  with  respect  to  his  system,  0'  will  find  experi- 

*  A  similar  invariance  of  electric  charge  has  been  made  fundamental  in  the 
author's  development  of  the  theory  of  similitude  (i.  e.,  the  theory  of  the  relativity 
of  size).  See  for  example  Phys.  Rev.,  vol.  3,  p.  244  (1914). 


172  Chapter  Twelve. 

mentally  for  a  similar  test  charge  that  moves  along  with  him  a  value 
for  the  force  Qe',  where  e'  can  be  calculated  from  with  the  help  of 
these  equations  (227).  Similar  remarks  would  apply  to  the  forces 
which  would  act  on  magnetic  poles. 

These  considerations  show  us  that  we  should  now  use  caution  in 
speaking  of  a  pure  electrostatic  or  pure  magnetic  field,  since  the 
description  of  an  electromagnetic  field  is  determined  by  the  particular 
choice  of  coordinates  with  reference  to  which  the  field  is  measured. 

159.  Nature  of  Electromotive  Force.     We  also  see  that  the  "  elec- 
tromotive "  force  which  acts  on  a  charge  moving  through  a  magnetic 
field  finds  its  interpretation  as  an  "  electric  "  force  provided  we  make 
use  of  a  system  of  coordinates  which  are  themselves  stationary  with 
respect  to  the  charge.     Such  considerations  throw  light  on  such  ques- 
tions, for  example,  as  to  the  seat  of  the  "  electromotive  "  forces  in 
"  homopolar  "  electric  dynamos  where  there  is  relative  motion  of  a 
conductor  and  a  magnetic  field. 

Derivation  of  the  Fifth  Fundamental  Equation. 

160.  We  may  now  make  use  of  this  fact  that  the  forces  acting  on 
a  moving  charge  of  electricity  may  be  treated  as  purely  electrostatic, 
by  using  a  set  of  coordinates  which  are  themselves  moving  along  with 
the  charge,  to  derive  the  fifth  fundamental  equation  of  electromagnetic 
theory. 

Consider  an  electromagnetic  field  having  the  values  e  and  h  for 
the  electric  and  magnetic  field  strengths  at  some  particular  point. 
What  will  be  the  value  of  the  electromagnetic  force  f  acting  per 
unit  volume  on  a  charge  of  density  p  which  is  passing  through  the 
point  in  question  with  the  velocity  u? 

To  solve  the  problem  take  a  system  of  coordinates  S'  which  itself 
moves  with  the  same  velocity  as  the  charge,  for  convenience  letting 
the  X-axis  coincide  with  the  direction  of  the  motion  of  the  charge. 
Since  the  charge  of  electricity  is  stationary  with  respect  to  this  system, 
the  force  acting  on  it  as  measured  in  units  of  this  system  will  be  by 
definition  equal  to  the  product  of  the  charge  by  the  strength  of  the 
electric  field  as  it  appears  to  an  observer  in  this  system,  so  that  we  may 
write 

F'  =  Q'e', 


Electromagnetic  Theory.  173 

or 

F*'  =  Q'e,',        F.'  =  Q'e,',        F.'  =  Q'e,'. 

For  the  components  of  the  electrical  field  ex't  evf,  ez'}  we  have  just 
obtained  the  transformation  equations  (227),  while  in  our  earlier 
dynamical  considerations  in  Chapter  VI  we  obtained  transformation 
equations  (61),  (62),  and  (63)  for  the  components  of  force.  Sub- 
stituting above  and  bearing  in  mind  that  ux  =  V,  uv  =  uz  =  0,  and 
that  Qf  —  Q,  we  obtain  on  simplification 

Fx  =  Qex, 

*•*•  — 


which  in  vectorial  form  gives  us  the  equation 


or  for  the  force  per  unit  volume 

f  =  Pe  +    [uXh]*.  (226) 


This  is  the  well-known  fifth  fundamental  equation  of  the  Maxwell- 
Lorentz  theory  of  electromagnetism.  We  have  already  indicated  the 
method  by  which  it  could  be  derived  from  the  principle  of  least  action. 
This  derivation,  however,  from  the  transformation  equations,  provided 
by  the  theory  of  relativity,  is  particularly  simple  and  attractive. 

Difference  between  the  Ether  and  the  Relativity  Theories  of  Electro- 
magnetism. 

161.  In  spite  of  the  fact  that  we  have  now  found  five  equations 
which  can  be  used  as  a  basis  for  electromagnetic  theory  which  agree 
with  the  requirements  of  relativity  and  also  have  exactly  the  same 
form  as  the  five  fundamental  equations  used  by  L<orentz  in  building 
up  the  stationary  ether  theory,  it  .must  not  be  supposed  that  the 
relativity  and  ether  theories  of  electromagnetism  are  identical.  Al- 
though the  older  equations  have  exactly  the  same  form  as  the  ones 
which  we  shall  henceforth  use,  they  have  a  different  interpretation, 
since  our  equations  are  true  for  measurements  made  with  the  help 
of  any  non-accelerated  set  of  coordinates,  while  the  equations  of 


174  Chapter  Twelve. 

Lorentz  were,  in  the  first  instance,  supposed  to  be  true  only  for  mea- 
surements which  were  referred  to  a  set  of  coordinates  which  were 
stationary  with  respect  to  the  assumed  luminiferous  ether.  Suppose, 
for  example,  we  desire  to  calculate  with  the  help  of  equation  (226), 

t  = 

the  force  acting  on  a  charged  body  which  is  moving  with  the  velocity  u; 
we  must  note  that  for  the  stationary  ether  theory,  u  must  be  the 
velocity  of  the  charged  body  through  the  ether,  while  for  us  u  may  be 
taken  as  the  velocity  past  any  set  of  unaccelerated  coordinates,  pro- 
vided e  and  h  are  measured  with  reference  to  the  same  set  of  co- 
ordinates. It  will  be  readily  seen  that  such  an  extension  in  the  mean- 
ing of  the  fundamental  equations  is  an  important  simplification. 

162.  A  word  about  the  development  from  the  theory  of  a  stationary 
ether  to  our  present  theory  will  not  be  out  of  place.  When  it  was 
found  that  the  theory  of  a  stationary  ether  led  to  incorrect  con- 
clusions in  the  case  of  the  Michelson-Morley  experiment,  the  hypo- 
thesis was  advanced  by  Lorentz  and  Fitzgerald  that  the  failure  of  that 
experiment  to  show  any  motion  through  the  ether  was  due  to  a  con- 
traction of  the  apparatus  in  the  direction  of  its  motion  through  the 

I         v? 
ether  in  the  ratio  1  :  */i  _  —  .     Lorentz  then  showed  that  if  all  sys- 

»  c 

terns  should  be  thus  contracted  in  the  line  of  their  motion  through  the 
ether,  and  observers  moving  with  such  system  make  use  of  suitably 
contracted  meter  sticks  and  clocks  adjusted  to  give  what  Lorentz 
called  the  "  local  time,"  their  measurements  of  electromagnetic 
phenomena  could  be  described  by  a  set  of  equations  which  have 
nearly  the  same  form  as  the  original  four  field  equations  which  would 
be  used  by  a  stationary  observer.  It  will  be  seen  that  Lorentz  was 
thus  making  important  progress  towards  our  present  idea  of  the  com- 
plete relativity  of  motion.  The  final  step  could  not  be  taken,  however, 
without  abandoning  our  older  ideas  of  space  and  time  and  giving  up 
the  Galilean  transformation  equations  as  the  basis  of  kinematics. 
It  was  Einstein  who,  with  clearness  and  boldness  of  vision,  pointed 
out  that  the'  failure  of  the  Michelson-Morley  experiment,  and  all 
other  attempts  to  detect  motion  through  the  ether,  is  not  due  to  a 


Electromagnetic  Theory.  175 

fortuitous  compensation  of  effects  but  is  the  expression  of  an  important 
general  principle,  and  the  new  transformation  equations  for  kinematics 
to  which  he  was  led  have  not  only  provided  the  basis  for  an  exact 
transformation  of  the  field  equations  but  have  so  completely  revo- 
lutionized our  ideas  of  space  and  time  that  hardly  a  branch  of  science 
remains  unaffected. 

163.  With  regard  to  the  present  status  of  the  ether  in  scientific 
theory,  it  must  be  definitely  stated  that  this  concept  has  certainly 
lost  both  its  fundamentality  and  the  greater  part  of  its  usefulness, 
and  this  has  been  brought  about  by  a  gradual  process  which  has  only 
found  its  culmination  in  the  work  of  Einstein.  Since  the  earliest 
days  of  the  luminiferous  ether,  the  attempts  of  science  to  increase  the 
substantiality  of  this  medium  have  met  with  little  success.  Thus 
we  have  had  solid  elastic  ethers  of  most  extreme  tenuity,  and  ethers 
with  a  density  of  a  thousand  tons  per  cubic  millimeter;  we  have  had 
quasi-material  tubes  of  force  and  lines  of  force ;  we  have  had  vibratory 
gyrostatic  ethers  and  perfect  gases  of  zero  atomic  weight;  but  after 
every  debauch  of  model-making,  science  has  recognized  anew  that  a 
correct  mathematical  description  of  the  actual  phenomena  of  light 
propagation  is  superior  to  any  of  these  sublimated  material  media. 
Already  for  Lorentz  the  ether  had  been  reduced  to  the  bare  function 
of  providing  a  stationary  system  of  reference  for  the  measurement  of 
positions  and  velocities,  and  now  even  this  function  has  been  taken 
from  it  by  the  work  of  Einstein,  which  has  shown  that  any  unaccel- 
erated  system  of  reference  is  just  as  good  as  any  other. 

To  give  up  the  notion  of  an  ether  will  be  very  hard  for  many 
physicists,  in  particular  since  the  phenomena  of  the  interference  and 
polarization  of  light  are  so  easily  correlated  with  familiar  experience 
with  wave  motions  in  material  elastic  media.  Consideration  will 
show  us,  however,  that  by  giving  up  the  ether  we  have  done  nothing 
to  destroy  the  periodic  or  polarizable  nature  of  a  light  disturbance. 
When  a  plane  polarized  beam  of  light  is  passing  through  a  given 
point  in  space  we  merely  find  that  the  electric  and  magnetic  fields  at 
that  point  he  on  perpendiculars  to  the  direction  of  propagation  and 
undergo  regular  periodic  changes  in  magnitude.  There  is  no  need  of 
going  beyond  these  actual  experimental  facts  and  introducing  any 
hypothetical  medium.  It  is  just  as  simple,  indeed  simpler,  to  say 


176  Chapter  Twelve. 

that  the  electric  or  magnetic  field  has  a  certain  intensity  at  a  given 
point  in  space  as  to  speak  of  a  complicated  sort  of  strain  at  a  given 
point  in  an  assumed  ether. 

Applications  to  Electromagnetic  Theory. 

164.  The  significant  fact  that  the  fundamental  equations  of  the 
new  electromagnetic  theory  have  the  same  form  as  those  of  Lorentz 
makes  it  of  course  possible  to  retain  in  the  structure  of  modern  elec- 
trical theory  nearly  all  the  results  of  his  important  researches,  care 
being  taken  to  give  his  mathematical  equations  an  interpretation  in 
accordance  with  the  fundamental  ideas  of  the  theory  of  relativity.     It 
is,  however,  entirely  beyond  our  present  scope  to  make  any  presenta- 
tion of  electromagnetic  theory  as  a  whole,  and  in  the  following  para- 
graphs we  shall  confine  ourselves  to  the  proof  of  a  few  theorems  which 
can  be  handled  with  special  ease  and  directness  by  the  methods  intro- 
duced by  the  theory  of  relativity. 

165.  The  Electric  and  Magnetic  Fields  around  a  Moving  Charge. 
Our  transformation  equations  for  the  electromagnetic  field  make  it 
very  easy  to  derive  expressions  for  the  field  around  a  point  charge  in 
uniform  motion.     Consider  a  point  charge  Q  moving  with  the  velocity 
V.     For  convenience  consider  a  system  of  reference  S  such  that  Q  is 
moving  along  the  X-axis  and  at  the  instant  in  question,  t  =  0,  let  the 
charge  coincide  with  the  origin  of  coordinates  0.     We  desire  now  to 
calculate  the  values  of  electric  field  e  and  the  magnetic  field  h  at  any 
point  in  space  x,  y,  z. 

Consider  another  system  of  reference,  S't  which  moves  along  with 
the  same  velocity  as  the  charge  Q,  the  origin  of  coordinates  0',  and 
the  charge  always  coinciding  in  position.  Since  the  charge  is  sta- 
tionary with  respect  to  their  new  system  of  reference,  we  shall  have 
the  electric  field  at  any  point  x',  yf,  z'  in  this  system  given  by  the 
equations 


,, Qtf_ 

'V 


(x'  +  y'  + 


ez'  = 


Electromagnetic  Theory.  177 

while  the  magnetic  field  will  obviously  be  zero  for  measurements  made 
in  system  S',  giving  us 

h,'  =  0, 
h,'  =  0, 
h,'  =  0. 

Introducing  our  transformation  equations  (9),  (10)  and  (11)  for  x', 
y'  and  z'  and  our  transformation  equations  (227)  and  (228)  for  the 
electric  and  magnetic  fields  and  substituting  t  =  0,  we  obtain  for  the 
values  of  e  and  h  in  system  S  at  the  instant  when  the  charge  passes 
through  the  point  0, 


e* 


3/2 


+  *  +  *r 


8''2' 


V2 


+  y2  +  02)3;2 


-  *» 


I         (         V2\ 

or,  putting  s  for  the  important  quantity  *\lx2  +  (  1  -    —  1  (y2  +  z2) 

and  writing  the  equations  in  the  vectorial  form  where  we  put 

r  =  (xi  +  yj  +  0k), 

we  obtain  the  familiar  equations  for  the  field  around  a  point  charge 
13 


178  Chapter  Twelve. 

in  uniform  motion  with  the  velocity  u  =  V  in  the  X-direction 


('-?)' 

e  =  Q^-  (232) 


h  =  -  [u  X  e].*  (233) 

c 

166.  The   Energy   of   a   Moving   Electromagnetic   System.     Our 

transformation  equations  will  permit  us  to  obtain  a  very  important 
expression  for  the  energy  of  an  isolated  electromagnetic  system  in 
terms  of  the  velocity  of  the  system  and  the  energy  of  the  same  system 
as  it  appears  to  an  observer  who  is  moving  along  with  it. 

Consider  a  physical  system  surrounded  by  a  shell  which  is  im- 
permeable to  electromagnetic  radiation.  This  system  is  to  be  thought 
of  as  consisting  of  the  various  mechanical  parts,  electric  charges  and 
electromagnetic  fields  which  are  inside  of  the  impermeable  shell. 
The  system  is  free  in  space,  except  that  it  may  be  acted  on  by  external 
electromagnetic  fields,  and  its  energy  content  thus  be  changed. 

Let  us  now  equate  the  increase  in  the  energy  of  the  system  to  the 
work  done  by  the  action  of  the  external  field  on  the  electric  charges 
in  the  system.  Since  the  force  which  a  magnetic  field  exerts  on  a 
charge  is  at  right  angles  to  the  motion  of  the  charge  it  does  no  work 
and  we  need  to  consider  only  the  work  done  by  the  external  electric 
field  and  may  write  for  the  increase  in  the  energy  of  the  system 

AE  =  ffff  p(exux  +  eyUy  +  ezuz)dx  dy  dz  dt,  (234) 

where  the  integration  is  to  be  taken  over  the  total  volume  of  the 
system  and  over  any  time  interval  in  which  we  may  be  interested. 

Let  us  now  transform  this  expression  with  the  help  of  our  trans- 
formation equations  for  the  electric  field  (227)  for  electric  charge 
(229),  and  for  velocities  (14-15-16).  Noting  that  our  fundamental 
equations  for  kinematic  quantities  give  us  dx  dy  dz  dt  =  dxf  dy'  dz'  dt', 
we  obtain 

AE  =  K  ffff  P'(ex'ux'  +  ey'uyr  +  ez'uz')dx'  dy'  dz'  dt' 

+  *V  ffff  P'     «/  +  ~f*  hzf  -  —-  hy'     dx'  dy'  dz'  dt'. 


Electromagnetic  Theory.  179 

Consider  now  a  system  which  both  at  the  beginning  and  end  of  our 
time  interval  is  free  from  the  action  of  external  forces;  we  may  then 
rewrite  the  above  equation  for  this  special  case  in  the  form 

AE  =  K&E'  +  KV  f  2Fz'dt', 


where,  in  accordance  with  our  earlier  equation  (234)  ,  AE'  is  the  increase 
in  the  energy  of  the  system  as  it  appears  to  observer  0'  and  2FX' 
is  the  total  force  acting  on  the  system  in  X-direction  as  measured 
by  0'. 

The  restriction  that  the  system  shall  be  unacted  on  by  external 
forces  both  at  the  beginning  and  end  of  our  time  interval  is  necessary 
because  it  is  only  under  those  circumstances  that  an  integration 
between  two  values  of  t  can  be  considered  as  an  integration  between 
two  definite  values  of  t',  simultaneity  in  different  parts  of  the  system 
not  being  the  same  for  observers  0  and  0'. 

We  may  now  apply  this  equation  to  a  specially  interesting  case. 
Let  the  system  be  of  such  a  nature  that  we  can  speak  of  it  as  being 
at  rest  with  respect  to  S',  meaning  thereby  that  all  the  mechanical 
parts  have  low  velocities  with  respect  to  S'  and  that  their  center  of 
gravity  moves  permanently  along  with  S'.  Under  these  circum- 
stances we  may  evidently  put  J  2Fx'dt'  =  0  and  may  write  the 
above  equation  in  the  form 


[--*' 

V1"? 


or 

dE 


where  u  is  the  velocity  of  the  system,  and  E°  is  its  energy  as  measured 
by  an  observer  moving  along  with  it.  The  energy  of  a  system  which 
is  unacted  on  by  external  forces  is  thus  a  function  of  two  variables,  its 
energy  EQ  as  measured  by  an  observer  moving  along  with  the  system 
and  its  velocity  u. 


180  Chapter  Twelve. 

We  may  now  write 

E  =  -        =#o  +  <}>(u)  +  const., 


where  <f>(u)  represents  the  energy  of  the  system  which  depends  solely 
on  the  velocity  of  the  system  and  not  on  the  changes  in  its  EQ  values. 
(f>(u)  will  thus  evidently  be  the  kinetic  energy  of  the  mechanical  masses 
in  the  system  which  we  have  already  found  (82)  to  have  the  value 

'-=  —  m0c2  where  mo  is  to  be  taken  as  the  total  mass  of  the 

"  c2" 
mechanical  part  of  our  system  when  at  rest.     We  may  now  write 

E  =  — : (w0c2  +  EQ)  —  m0c2  +  const. 

u2 


Or,  assuming  as  before  that  the  constant  is  equal  to  m0c2,  which  will 
be  equivalent  to  making  a  system  which  has  zero  energy  also  have 
zero  mass,  we  obtain 


which  is  the  desired  expression  for  the  energy  of  an  isolated  system 
which  may  contain  both  electrical  and  mechanical  parts. 

167.  Relation  between  Mass  and  Energy.  This  expression  for  the 
energy  of  a  system  that  contains  electrical  parts  permits  us  to  show 
that  the  same  relation  which  we  found  between  mass  and  energy  for 
mechanical  systems  also  holds  in  the  case  of  electromagnetic  energy. 
Consider  a  system  containing  electromagnetic  energy  and  enclosed 
by  a  shell  which  is  impermeable  to  radiation.  Let  us  apply  a  force  F 
to  the  system  in  such  a  way  as  to  change  the  velocity  of  the  system 
without  changing  its  E0  value.  We  can  then  equate  the  work  done 
per  second  by  the  force  to  the  rate  of  increase  of  the  energy  of  the 
system.  We  have 

dE 


Electromagnetic  Theory. 


181 


But  from  equation  (235)  we  can  obtain  a  value  for  the  rate  of  increase 

dE 
of  energy  -j-  ,  giving  us 


F-u  =  Fxux  +  FyUy  +  F2uz  =  (  m0  +-j 
and  solving  this  equation  for  F  we  obtain 

F  =  dt 
which  for  low  velocities  assumes  the  form 


du 

fit 


(236) 


(237) 


Examination  of  these  expressions  shows  that  our  system  which 
contains  electromagnetic  energy  behaves  like  an  ordinary  mechanical 


(n     ,Et\ 
(™»  +  ^) 


—  J    at  low  velocities  or    m0  +  — 

—7-  at 


any  desired  velocity  w.     To  the  energy  of  the  system  EQ,  part  of  which 


is  electromagnetic,  we  must  ascribe  the  mass  —  just  as  we  found  in 

the  case  of  mechanical  energy.     We  realize  again  that  matter  and 
energy  are  but  different  names  for  the  same  fundamental  entity, 

1021  ergs  of  energy  having  the  mass  1  gram. 

/ 

The  Theory  of  Moving  Dielectrics. 

168.  The  principle  of  relativity  proves  to  be  very  useful  for  the 
development  of  the  theory  of  moving  dielectrics. 

It  was  first  shown  by  Maxwell  that  a  theory  of  electromagnetic 
phenomena  in  material  media  can  be  based  on  a  set  of  field  equations 
similar  in  form  to  those  for  free  space,  provided  we  introduce  besides 
the  electric  and  magnetic  field  strengths,  E  and  F,  two  new  field  vectors > 


182  Chapter  Twelve. 

the  dielectric  displacement  D  and  the  magnetic  induction  B,  and 
also  the  density  of  electric  current  in  the  medium  i.  These  quantities 
are  found  to  be  connected  by  the  four  following  equations  similar  in 
form  to  the  four  field  equations  for  free  space: 


curl]!  =  -(  — +  i  ),  (238) 

1  dB 

curl  E  =   -  -  —  ,  (239) 

C    ut 

div  D  =  p,  (240) 

div   B  =  0.  (241) 

For  stationary  homogeneous  media,  the  dielectric  displacement, 
magnetic  induction  and  electric  current  are  connected  with  the 
electric  and  magnetic  field  strengths  by  the  following  equations: 

D  =  eE,  (242) 

B  -  MH,  (243) 

i  =  crE,  (244) 

where  e  is  the  dielectric  constant,  //  the  magnetic  permeability  and  a 
the  electrical  conductivity  of  the  medium  in  question. 

169.  Relation  between  Field  Equations  for  Material  Media  and 
Electron  Theory.  It  must  not  be  supposed  that  the  four  field  equa- 
tions (238-241)  for  electromagnetic  phenomena  in  material  media  are 
in  any  sense  contradictory  to  the  four  equations  (221-224)  for  free 
space  which  we  took  as  the  fundamental  basis  for  our  development  of 
electromagnetic  theory.  As  a  matter  of  fact,  one  of  the  main  achieve- 
ments of  modern  electron  theory  has  been  to  show  that  the  electro- 
magnetic behavior  of  material  media  can  be  explained  in  terms  of 
the  behavior  of  the  individual  electrons  and  ions  which  they  contain, 
these  electrons  and  ions  acting  in  accordance  with  the  four  fundamental 
field  equations  for  free  space.  Thus  our  new  equations  for  material 
media  merely  express  from  a  macroscopic  point  of  view  the  statistical 
result  of  the  behavior  of  the  individual  electrons  in  the  material  in 
question.  E  and  H  in  these  new  equations  are  to  be  looked  upon  as 
the  average  values  of  e  and  h  which  arise  from  the  action  of  the 
individual  electrons  in  the  material,  the  process  of  averaging  being  so 


Electromagnetic  Theory.  183 

carried  out  that  the  results  give  the  values  which  a  macroscopic  ob- 
server would  actually  find  for  the  electric  and  magnetic  forces  acting 
respectively  on  a  unit  charge  and  a  unit  pole  at  the  point  in  question. 
These  average  values,  E  and  H,  will  thus  pay  no  attention  to  the 
rapid  fluctuations  of  e  and  h  which  arise  from  the  action  and  motion 
of  the  individual  electrons,  the  macroscopic  observer  using  in  fact 
differentials  for  time,  dt,  and  space,  dx,  which  would  be  large  from  a 
microscopic  or  molecular  viewpoint. 

Since  from  a  microscopic  point  of  view  E  and  H  are  not  really 
the  instantaneous  values  of  the  field  strength  at  an  actual  point  in 
space,  it  has  been  found  necessary  to  introduce  two  new  vectors, 
electric  displacement,  D,  and  magnetic  induction,  B,  whose  time 
rate  of  change  will  determine  the  curl  of  E  and  H  respectively.  It  will 
evidently  be  possible,  however,  to  relate  D  and  B  to  the  actual  electric 
and  magnetic  fields  e  and  h  produced  by  the  individual  electrons, 
and  this  relation  has  been  one  of  the  problems  solved  by  modern 
electron  theory,  and  the  field  equations  (238-241)  for  material  media 
have  thus  been  shown  to  stand  in  complete  agreement  with  the  most 
modern  views  as  to  the  structure  of  matter  and  electricity.  For 
the  purposes  of  the  rest  of  our  discussion  we  shall  merely  take  these 
equations  as  expressing  the  experimental  facts  in  stationary  or  in 
moving  media. 

170.  Transformation  Equations  for  Moving  Media.  Since  equa- 
tions (238  to  241)  are  assumed  to  give  a  correct  description  of  electro- 
magnetic phenomena  in  media  whether  stationary  or  moving  with 
respect  to  our  reference  system  S,  it  is  evident  that  the  equations 
must  be  unchanged  in  form  if  we  refer  our  measurements  to  a  new 
system  of  coordinates  S'  moving  past  S,  say,  with  the  velocity  V  in  the 
X-direction. 

As  a  matter  of  fact,  equations  (238  to  241)  can  be  transformed 
into  an  entirely  similar  set 


curl  E'  =  -  -  ~ 
c    dt 

div  D'  =  p', 
div   B'  =  0, 


184  Chapter  Twelve. 

provided  we  substitute  for  x,  y,  z  and  t  the  values  of  xf,  y',  z'  and  tr 
given  by  the  fundamental  transformation  equations  for  space  and 
time  (9  to  12),  and  substitute  for  the  other  quantities  in  question  the 
relations 

Ex'  =  Et, 

E'  = 


'        D  (245) 

=  Lfx, 


D;  = 


fly  - 

(246) 


B/« 

B,'  = 

P'  =  <(P-£;,), 

(247) 


It  will  be  noted  that  for  free  space  these  equations  will  reduce  to 
the  same  form  as  our  earlier  transformation  equations  (227  to  229) 
since  we  shall  have  the  simplifications  D  =  E,  B  =  H  and  i  =  pu. 

We  may  also  call  attention  at  this  point  to  the  fact  that  our  funda- 


Electromagnetic  Theory.  185 

mental  equations  for  electromagnetic  phenomena  (238-241)  in  di- 
electric media  might  have  been  derived  from  the  principle  of  least 
action,  making  use  of  an  expression  for  kinetic  potential  which  could 

//E-D      H-B\ 
dV(  —  -  ---  9/>  anc*  ^  w^  ke  noticed 

that  our  transformation  equations  for  these  quantities  are  such  as  to 

TT 

preserve  that  necessary  invariance  for  —=    =  which  we  found  in 


Chapter  IX  to  be  the  general  requirement  for  any  dynamical,  develop- 
ment which  agrees  with  the  theory  of  relativity. 

171.  We  are  now  in  a  position  to  handle  the  theory  of  moving 
media.  Consider  a  homogeneous  medium  moving  past  a  system  of 
coordinates  S  in  the  X-direction  with  the  velocity  F;  our  problem  is 
to  discover  relations  between  the  various  electric  and  magnetic 
vectors  in  this  medium.  To  do  this,  consider  a  new  system  of  co- 
ordinates S'  also  moving  past  our  original  system  with  the  velocity  V. 
Since  the  medium  is  stationary  with  respect  to  this  new  system  S'  we 
may  write  for  measurements  referred  to  S'  in  accordance  with  equa- 
tions (242  to  244)  the  relations 

D'  =  eE', 

B'  =  0H', 

i'  =  <rE', 

which,  as  we  have  already  pointed  out,  are  known  experimentally  to 
be  true  in  the  case  of  stationary,  homogeneous  media,  e,  p  and  a-  are 
evidently  the  values  of  dielectric  constant,  permeability  and  con- 
ductivity of  the  material  in  question,  which  would  be  found  by  an 
experimenter  with  respect  to  whom  the  medium  is  stationary. 

Making  use  of  our  transformation  equations  (245  to  247)  we  can 
obtain  by  obvious  substitutions  the  following  set  of  relations  for 
measurements  made  with  respect  to  the  original  system  of  coordi- 
nates S: 


(248) 


186 


Chapter  Twelve. 


Bx  = 

V 
BV+-EZ  = 


(249) 


i  -  -Bz\, 


(250) 


z  = 


172.  Theory  of  the  Wilson  Experiment.  The  equations  which  we 
have  just  developed  for  moving  media  are,  as  a  matter  of  fact,  in 
complete  accord  with  the  celebrated  experiment  of  H.  A.  Wilson  on 
moving  dielectrics  and  indeed  all  other  experiments  that  have  been 
performed  on  moving  media. 

Wilson's  experiment  consisted  in  the  rotation  of  a  hollow  cylinder 
of  dielectric,  in  a  magnetic  field  which  was  parallel  to  the  axis  of  the 
cylinder.  The  inner  and  outer  surfaces  of  the  cylinder  were  covered 
with  a  thin  metal  coating,  and  arrangements  made  with  the  help  of 
wire  brushes  so  that  electrical  contact  could  be  made  from  these 
coatings  to  the  pairs  of  quadrants  of  an  electrometer.  By  reversing 
the  magnetic  field  while  the  apparatus  was  in  rotation  it  was  possible 
to  measure  with  the  electrometer  the  charge  produced  by  the  electrical 
displacement  in  the  dielectric.  We  may  make  use  of  our  equations 
to  compute  the  quantitative  size  of  the  effect. 

z 


FIG.  15. 


Electromagnetic  Theory.  187 

Let  figure  15  represent  a  cross-section  of  the  rotating  cylinder. 
Consider  a  section  of  the  dielectric  A  A  which  is  moving  perpendicularly 
to  the  plane  of  the  paper  in  the  X-direction  with  the  velocity  V.  Let 
the  magnetic  field  be  in  the  F-direction  parallel  to  the  axis  of  rotation. 
The  problem  is  to  calculate  dielectric  displacement  Dz  in  the  Z- 
direction. 

Referring  to  equations  (248)  we  have 

*.+!*.•.  .(*.+£«.), 

and,  substituting  the  value  of  Bv  given  by  equations  (249), 


we  obtain 


(yt 
-t 
€M  c2 


V 

or,  neglecting  terms  of  orders  higher  than  —  ,  we  have 

V 

Dz  =  eEz  +  -  (eju  -  l)Hv.  (251) 

For  a  substance  whose  permeability  is  practically  unity  such  as 
Wilson  actually  used  the  equation  reduces  to 

D2  =  eEz  +  -  (€  -  l)#y, 
c 

and  this  was  found  to  fit  the  experimental  facts,  since  measurements 
with  the  electrometer  show  the  surface  charge  actually  to  have  the 
magnitude  Dz  per  square  centimeter  in  accordance  with  our  equation 
div  D  =  p. 

It  would  be  a  matter  of  great  interest  to  repeat  the  Wilson  experi- 
ment with  a  dielectric  of  high  permeability  so  that  we  could  test  the 
complete  equation  (251).  This  is  of  some  importance  since  the 
original  Lorentz  theory  led  to  a  different  equation, 

V 

D,  =  *EZ  +  -  (e  - 


CHAPTER  XIII. 
FOUR-DIMENSIONAL  ANALYSIS. 

173.  In  the  present  chapter  we  shall  present  a  four-dimensional 
method  of  expressing  the  results  of  the  Einstein  theory  of  relativity, 
a  method  which  was  first  introduced  by  Minkowski,  and  in  the  form 
which  we  shall  use,  principally  developed  by  Wilson  and  Lewis.     The 
point  of  view  adopted,  consists  essentially  in  considering  the  properties 
of  an  assumed  four-dimensional  space  in  which  intervals  of  time  are 
thought  of  as  plotted  along  an. axis  perpendicular  to  the  three  Car- 
tesian axes  of  ordinary  space,  the  science  of  kinematics  thus  becoming 
the  geometry  of  this  new  four-dimensional  space. 

The  method  often  has  very  great  advantages  not  only  because  it 
sometimes  leads  to  considerable  simplification  of  the  mathematical 
form  in  which  the  results  of  the  theory  of  relativity  are  expressed, 
but  also  because  the  analogies  between  ordinary  geometry  and  the 
geometry  of  this  imaginary  space  often  suggest  valuable  modes  of 
attack.  On  the  other  hand,  in  order  to  carry  out  actual  numerical 
calculations  and  often  in  order  to  appreciate  the  physical  significance 
of  the  conclusions  arrived  .at,  it  is  necessary  to  retranslate  the  results 
obtained  by  this  four-dimensional  method  into  the  language  of  ordinary 
kinematics.  It  must  further  be  noted,  moreover,  that  many  im- 
portant results  of  the  theory  of  relativity  can  be  more  easily  obtained 
if  we  do  not  try  to  employ  this  four-dimensional  geometry.  The 
reader  should  also  be  on  his  guard  against  the  fallacy  of  thinking  that 
extension  in  time  is  of  the  same  nature  as  extension  in  space  merely 
because  intervals  of  space  and  time  can  both  be  represented  by 
plotting  along  axes  drawn  on  the  same  piece  of  paper. 

174.  Idea  of  a  Time  Axis.     In  order  to  grasp  the  method  let  us 
consider  a  particle  constrained  to  move  along  a  single  axis,  say  OX, 
and  let  us  consider  a  time  axis  OT  perpendicular  to  OX.     Then  the 
position  of  the  particle  at  any  instant  of  time  can  be  represented  by  a 
point  in  the  XT  plane,  and  its  motion  as  time  progresses  by  a  line  in 
the  plane.     If,  for  example,  the  particle  were  stationary,  its  behavior 

188 


Four  Dimensional  Analysis. 


189 


in  time  and  space  could  be  represented  by  a  line  parallel  to  the  time 
axis  OT  as  shown  for  example  by  the  line  ab  in  figure  16.      A  particle 


/ 

I  / 


/ 


-sr 


/ 


FIG.  16. 


dx 


moving  with  the  uniform  velocity  u  =  ^-  could  be  represented  by  a 

straight  line  ac  making  an  angle  with  the  time  axes,  and  the  kine- 
matical  behavior  of  an  accelerated  particle  could  be  represented  by  a 
curved  line. 

By  conceiving  of  a  /owr-dimensional  space  we  can  extend  this 
method  which  we  have  just  outlined  to  include  motion  parallel  to 
all  three  space  axes,  and  in  accordance  with  the  nomenclature  of 
Minkowski  might  call  such  a  geometrical  representation  of  the  space- 
time  manifold  "  the  world,"  and  speak  of  the  points  and  lines  which 
represent  the  instantaneous  positions  and  the  motions  of  particles  as 
"  world-points  "  and  "  world-lines." 

175.  Non-Euclidean  Character  of  the  Space.  It  will  be  at  once 
evident  that  the  graphical  method  of  representing  kinematical  events 
which  is  shown  by  Figure  16  still  leaves  something  to  be  desired.  One 
of  the  most  important  conclusions  drawn  from  the  theory  of  relativity 
was  the  fact  that  it  is  impossible  for  a  particle  to  move  with  a  velocity 
greater  than  that  of  light,  and  it  is  evident  that  there  is  nothing  in 
our  plot  to  indicate  that  fact,  since  we  could  draw  a  line  making  any 
desired  angle  with  the  time  axis,  up  to  perpendicularity,  and  thus 


190 


Chapter  Thirteen. 


represent  particles  moving  with  any  velocity  up  to  infinity, 

Ax 


u  = 


At 


It  is  also  evident  that  there  is  nothing  in  our  plot  to  correspond  to 
that  invariance  in  the  velocity  of  light  which  is  a  cornerstone  of  the 
theory  of  relativity.  Suppose,  for  example,  the  line  OC,  in  figure  17, 


0 


1 


I 


I 


/I 
i 
i 

i 

l 


,'C 


FIG.  17. 


Ax 


represents  the  trajectory  of  a  beam  of  light  with  the  velocity  —   =  c; 

there  is  then  nothing  so  far  introduced  into  our  method  of  plotting 
to  indicate  the  fact  that  we  could  not  equally  well  make  use  of  another 
set  of  axes  OX'T',  inclined  to  the  first  and  thus  giving  quite  a  different 

Axf 
value,  —7 ,  to  the  velocity  of  the  beam  of  light. 

There  are  a  number  of  methods  of  meeting  this  difficulty  and 
obtaining  the  invariance  for  the  four-dimensional  expression  x2  +  y2 
+  z2  —  cH2  (see  Chapter  IV)  which  must  characterize  our  system  of 
kinematics.  One  of  these  is  to  conceive  of  a  four-dimensional  Eu- 


Four  Dimensional  Analysis.  191 

clidean  space  with  an  imaginary  time  axis,  such  that  instead  of  plotting 
real  instants  in  time  along  this  axis  we  should  plot  the  quantity 
I  =  id  where  i  =  V  —  1.  In  this  way  we  should  obtain  in  variance 
for  the  quantity  x2  +  y2  +  z2  +  I2  =  x2  +  y2  +  z2  -  c2t2,  since  it  may 
be  regarded  as  the  square  of  the  magnitude  of  an  imaginary  four- 
dimensional  radius  vector.  This  method  of  treatment  has  been 
especially  developed  by  Minkowski,  Laue,  and  Sommerfeld.  Another 
method  of  attack,  which  has  been  developed  by  Wilson  and  Lewis 
and  is  the  one  whrch  we  shall  adopt  in  this  chapter,  is  to  use  a  real 
time  axis,  for  plotting  the  real  quantity  ct,  but  to  make  use  of  a  non- 
Euclidean  four-dimensional  space  in  which  the  quantity  (x2  +  y2  +  z2 
-  c2t2)  is  itself  taken  as  the  square  of  the  magnitude  of  a  radius  vector. 
This  latter  method  has  of  course  the  disadvantages  that  come  from 
using  a  non-Euclidean  space;  we  shall  find,  however,  that  these  reduce 
largely  to  the  introduction  of  certain  rules  as  to  signs.  The  method 
has  the  considerable  advantage  of  retaining  a  real  time  axis  which  is 
of  some  importance,  if  we  wish  to  visualize  the  methods  of  attack  and 
to  represent  them  graphically. 

We  may  now  proceed  to  develop  an  analysis  for  this  non-Euclidean 
space.  We  shall  find  this  to  be  quite  a  lengthy  process  but  at  its 
completion  we  shall  have  a  very  valuable  instrument  for  expressing 
in  condensed  language  the  results  of  the  theory  of  relativity.  Our 
method  of  treatment  will  be  almost  wholly  analytical,  and  the  geo- 
metrical analogies  may  be  regarded  merely  as  furnishing  convenient 
names  for  useful  analytical  expressions.  A  more  geometrical  method 
of  attack  will  be  found  in  the  original  work  of  Wilson  and  Lewis. 

PART  I.     VECTOR  ANALYSIS  OF  THE  NON-EUCLIDEAN  FOUR- 

DIMENSIONAL  MANIFOLD. 

i 

176.  Consider  a  four-dimensional  manifold  in  which  the  position 
of  a  point  is  determined  by  a  radius  vector 


r  =  (xiki  +  z2k2  + 

where  ki,  k2,  k3  and  k4  may  be  regarded  as  unit  vectors  along  four 
mutually  perpendicular  axes  and  x\,  xz,  x3)  and  x4  as  the  magnitudes 
of  the  four  components  of  r  along  these  four  axes.  We  may  identify 
xi,  x>2,  and  z3  with  the  three  spatial  coordinates  of  a  point  x,  y  and  z 


192  Chapter  Thirteen. 

with  reference  to  an  ordinary  set  of  space  axes  and  consider  #4  as  a 
coordinate  which  specifies  the  time  (multiplied  by  the  velocity  of 
light)  when  the  occurrence  in  question  takes  place  at  the  point  xyz. 
We  have 

xi  =  x,         x%  =  y,         xs  =  z,        x±  =  ct,  (252) 

and  from  time  to  time  we  shall  make  these  substitutions  when  we 
wish  to  interpret  our  results  in  the  language  of  ordinary  kinematics. 
We  shall  retain  the  symbols  Xi,  x2,  x3,  and  x\  throughout  our  develop- 
ment, however,  for  the  sake  of  symmetry. 

177.  Space,  Time  and  Singular  Vectors.  Our  space  will  differ  in 
an  important  way  from  Euclidean  space  since  we  shall  consider  three 
classes  of  one-vector,  space,  time  and  singular  vectors.  Considering 
the  coordinates  rci,  x%,  x$  and  x4  which  determine  the  end  of  a  radius 
vector, 
Space  or  y-vectors  will  have  components  such  that 


and  we  shall  put  for  their  magnitude 


s  =  Vzi2  +  x?  +  z32  -  z42.  (253) 

Time  or  5-vectors  will  have  components  such  that 

and  we  shall  put  for  their  magnitude 

-       -V/y.   2  /v.    2     /y    2  /y»2  f  9  ^4.^ 

S    —      \2/4     —   *Ci     —   J/2     —   «£3  •  k^***/ 

Singular  or  a-vectors  will  have  components  such  that 

and  their  magnitude  will  be  zero. 

178.  Invariance  of  x2  +  y2  +  z2  —  c2*2.  Since  we  shall  naturally 
consider  the  magnitude  of  a  vector  to  be  independent  of  any  particular 
choice  of  axes  we  have  obtained  at  once  by  our  definition  of  magnitude 
for  any  rotation  of  axes  that  invariance  for  the  expression 

(zi2  +  z22  +  za2  —  ^42)  =  (x2  +  y2  +  z2  - 


Four  Dimensional  Analysis.  193 

which  is  characteristic  of  the  Lorentz  transformation,  and  have  thus 
evidently  set  up  an  imaginary  space  which  will  be  suitable  for  plotting 
kinematical  events  in  accordance  with  the  requirements  of  the  theory 
of  the  relativity  of  motion. 

179.  Inner  Product  of  One-Vectors.  We  shall  define  the  inner 
product  of  two  one-vectors  with  the  help  of  the  following  rules  for  the 
multiplication  of  unit  vectors  along  the  axes 

krki  =  k2-ks  =  ka-ka  =  1,         k^k*  =  --  1,         kn-km  =  0.    (255) 

It  should  be  noted,  of  course,  that  there  is  no  particular  sig- 
nificance in  picking  out  the  product  krki  as  the  one  which  is  nega- 
tive; it  would  be  equally  possible  to  develop  a  system  in  which  the 
products  ki  •  ki,  k2  •  k2,  and  k3  •  ks  should  be  negative  and  kj  •  kt  positive. 

The  above  rules  for  unit  vectors  are  sufficient  to  define  completely 
the  inner  product  provided  we  include  the  further  requirements  that 
this  product  shall  obey  the  associative  law  for  a  scalar  factor  and  the 
distributive  and  commutative  laws,  namely 

(na)-b  =  n(a-b)  =  (a-b)(n), 

a.(b  +  c)  =  a-b  +  a-c,  (256) 

a-b  =  b-a. 

For  the  inner  product  of  a  one-vector  by  itself  we  shah1  have,  in 
accordance  with  these  rules, 


r-  r  = 

(257) 


and  hence  may  use  the  following  expressions  for  the  magnitudes  of 
vectors  in  terms  of  inner  product 

s  —  Vr-r  for  ^-vectors,        s  =  V—  r-r  for  5-vectors.     (258) 

For  curved  lines  we  shall  define  interval  along  the  curve  by  the 
equations 

J  ds  =  J  Vdr  •  dr  for  7-curves, 

_  (259) 

J  ds  =  J  V—  dr-dr  for  6-curves. 
14 


194  Chapter  Thirteen. 

Our  rules  further  show  us  that  we  may  obtain  the  space  components 
of  any  one  vector  by  taking  its  inner  product  with  a  unit  vector 
along  the  desired  axis  and  may  obtain  the  time  component  by  taking 
the  negative  of  the  corresponding  product.  Thus 


r-k2  =  (siki  +  z2k2  -j-  z3k3  +  z4k4)  -k2  =  xz, 

(260) 
r-k3  = 

r-k4  = 

We  see  finally  moreover  in  general  that  the  inner  product  of  any 
pair  of  vectors  will  be  numerically  equal  to  the  product  of  the  mag- 
nitude of  either  by  the  projection  of  the  other  upon  it,  the  sign  de- 
pending on  the  nature  of  the  vectors  involved. 

180.  Non-Euclidean  Angle.     We  shall  define  the  non-Euclidean 
angle  6  between  two  vectors  rx  and  r2  in  terms  of  their  magnitudes 
Si  and  s2  by  the  expressions 

d=  Ti-r2  =  (si  X  projection  s2)  =  SiS2  cosh  0,  (261) 

the  sign  depending  on  the  nature  of  the  vectors  in  the  way  indicated 
in  the  preceding  section.  We  note  the  analogy  between  this  equation 
and  those  familiar  in  Euclidean  vector-analysis,  the  hyperbolic 
trigonometeric  functions  taking  the  place  of  the  circular  functions 
used  in  the  more  familiar  analysis. 

For  the  angle  between  unit  vectors  k  and  k7  we  shall  have 

cosh  0  =  =t  k-kr,  (262) 

where  the  sign  must  be  chosen  so  as  to  make  cosh  6  positive,  the 
plus  sign  holding  if  both  are  7-vectors  and  the  minus  sign  if  both  are 
5-vectors. 

181.  Kinematical  Interpretation  of  Angle  in  Terms  of  Velocity. 
At  this  point  we  may  temporarily  interrupt  the  development  of  our 
four-dimensional  analysis  to  consider  a  kinematical  interpretation  of 
non-Euclidean  angles  in  terms  of  velocity.     It  will  be  evident  from 
our  introduction  that  the  behavior  of  a  moving  particle  can  be  repre- 
sented in  our  four-dimensional  space  by  a  6-curve,*  each  point  on 

*  It  is  to  be  noted  that  the  actual  trajectories  of  particles  are  all  of  them  repre- 
sented by  5-curves  since  as  we  shall  see  ^-curves,  would  correspond  to  velocities 
greater  than  that  of  light. 


Four  Dimensional  Analysis.  195 

this  curve  denoting  the  position  of  the  particle  at  a  given  instant  of 
time,  and  it  is  evident  that  the  velocity  of  the  particle  will  be  deter- 
mined by  the  angle  which  this  curve  makes  with  the  axes. 

Let  r  be  the  radius  vector  to  a  given  point  on  the  curve  and  con- 
sider the  derivative  of  r  with  respect  to  the  interval  s  along  the  curve; 
we  have 

dr      dxi  dxz,         dx^  dx* 

w  =  d-s=^k'+^k2  +  dFk°  +  dFk"  (263) 

and  this  may  be  regarded  as  a  unit  vector  tangent  to  the  curve  at  the 
point  in  question. 

If  0  is  the  angle  between  the  ki  axis  and  the  tangent  to  the  curve 
at  the  point  in  question,  we  have  by  equation  (262) 

dx4 
cosh  </>  =  -  w-k4  =  -^  ; 

making  the  substitutions  for  xi,  x2,  x3,  and  XA,  in  terms  of  x}  y,  z  and  t 
we  may  write,  however, 


ds  =     dz42  -  dx?  -  dxj  -  dxj  =      l  -       cdt,  (264) 

which  gives  us 

cosh  0  =     ,  (265) 


and  by  the  principles  of  hyperbolic  trigonometry  we  may  write  the 
further  relations 

u 

L=,  (266) 


tanh  0  =  -  .  (267) 

c 

VECTORS  OF  HIGHER  DIMENSIONS 

182.  Outer  Products.     We  shall  define  the  outer  product  of  two 
one-vectors  so  that  it  obeys  the  associative  law  for  a  scalar  factor,  the 


196  Chapter  Thirteen. 

distributive  law  and  the  anti-commutative  law,  namely, 

(na)  X  b  =  n(a  X  b)  =  a  X  (wb), 

a  X  (b  +  c)  =aXb+aXc      (a-f-b)Xc  =  aXc  +  bXc;    (268) 
a  x  b  =  -bXa. 

From  a  geometrical  point  of  view,  we  shall  consider  the  outer 
product  of  two  one-vectors  to  be  itself  a  two-vector,  namely  the  paral- 
lelogram, or  more  generally,  the  area  which  they  determine.  The 
sign  of  the  two-vector  may  be  taken  to  indicate  the  direction  of  pro- 
gression clockwise  or  anti-clockwise  around  the  periphery.  In  order 
to  accord  with  the  requirement  that  the  area  of  a  parallelogram  deter- 
mined by  two  lines  becomes  zero  when  they  are  rotated  into  the  same 
direction,  we  may  complete  our  definition  of  outer  product  by  adding 
the  requirement  that  the  outer  product  of  a  vector  by  itself  shall  be 
zero. 

a  X  a  =  0.  (269) 

We  may  represent  the  outer  products  of  unit  vectors  along  the 
chosen  axes  as  follows: 

kx  X  ki  =  k2  X  k2  =  k3  X  k3  =  k4  X  k4  =  0, 

ki  X  k2  =  -  k2  X  ki  =  kia  =  -  k2i,  (270) 

ki  X  k3  =  —  k3  X  ki  =  kis  =  —  k3i,     etc., 

where  we  may  regard  k]2,  for  example,  as  a  unit  parallelogram  in  the 
plane  XiOX*. 

We  shall  continue  to  use  small  letters  in  Clarendon  type  for  one- 
vectors  and  shall  use  capital  letters  in  Clarendon  type  for  two-vectors. 
The  components  of  a  two-vector  along  the  six  mutually  perpendicular 
planes  XiOXz,  XiOX3,  etc.,  may  be  obtained  by  expressing  the  one- 
vectors  involved  in  terms  of  their  components  along  the  axes  and 
carrying  out  the  indicated  multiplication,  thus: 


A  =  a  X  b  =  (aiki  +  a2k2  +  a3k3  +  a4k4) 
X  (fciki  +  62k2  +  &3k3  +  64k4) 


(271) 
—  a362)k23  +  («2&4  —  a462)k24  +  (a364  —  a463)k34, 


Four  Dimensional  Analysis.  197 

or,  calling  the  quantities  (a  162  —  «2&i),  etc.,  the  component  magni- 
tudes of  A,  A  12,  etc.,  we  may  write 

A  =  Ai2k12  +  A13k13  +  4i4kI4  +  A23k23  +  A24k24  +  A34kM.    (272) 


concept  of  outer  product  may  be  extended  to  include  the 
idea  of  vectors  of  higher  number  of  dimensions  than  two.  Thus  the 
outer  product  of  three  one-vectors,  or  of  a  one-vector  and  a  two-vector 
will  be  a  three-vector  which  may  be  regarded  as  a  directed  parallele- 
piped in  our  four-dimensional  space.  The  outer  product  of  four  one- 
vectors  will  lead  to  a  four-dimensional  solid  .which  would  have  direction 
only  in  a  space  of  more  than  four  dimensions  and  hence  in  our  case 
will  be  called  a  pseudo-scalar.  The  outer  product  of  vectors  the 
sum  of  whose  dimensions  is  greater  than  that  of  the  space  considered 
will  vanish. 

The  results  which  may  be  obtained  from  different  types  of  outer 
multiplication  are  tabulated  below,  where  one-vectors  are  denoted 
by  small  Clarendon  type,  two-vectors  by  capital  Clarendon  type, 
three-vectors  by  Tudor  black  capitals,  and  pseudo-scalars  by  bold  face 
Greek  letters. 


A  =  a  X  b  =  —  b  X  a  =  («i62  —  a26i)ki2  +  (a  164  —  a36i)ki3 
+  (a  164  -  a4&i)ki4  +  (^263  -  0362)  k23  +  (a264  -  a462)k21 
+  (a364  -  a463)k34, 

H  =  c  X  A  =  (ciA23  -  c2Ais  +  c3A12)k123 


(273) 
4.23234, 

a-d.XH---HXd 

=  (d&tu  —  d22(i34 
a  =  A  X  B  =  G4i2B34  -  ^13524  +  AUB 


The  signs  in  these  expressions  are  determined  by  the  general  rule 
that  the  sign  of  any  unit  vector  knmo  will  be  reversed  by  each  transposition 
of  the  order  of  a  pair  of  adjacent  subscripts,  thus  : 

=  head,         etc.,   •  •  -.  (274) 


198  Chapter  Thirteen. 

183.  Inner  Product  of  Vectors  in  General.  We  have  previously 
defined  the  inner  product  for  the  special  case  of  a  pair  of  one-vectors, 
in  order  to  bring  out  some  of  the  important  characteristics  of  our 
non-Euclidean  space.  We  may  now  give  a  general  rule  for  the  inner 
product  of  vectors  of  any  number  of  dimensions. 

The  inner  product  of  any  pair  of  vectors  follows  the  associative 
law  for  scalar  factors,  and  follows  the  distributive  and  commutative 
laws. 

Since  we  can  express  any  vector  in  terms  of  its  components,  the 
above  rules  will  completely  determine  the  inner  product  of  any  pair 
of  vectors  provided  that  we  also  have  a  rule  for  obtaining  the  inner 
products  of  the  unit  vectors  determined  by  the  mutually  perpendicular 
axes.  This  rule  is  as  follows:  Transpose  the  subscripts  of  the  unit 
vectors  involved  so  that  the  common  subscripts  occur  at  the  end  and 
in  the  same  order  and  cancel  these  common  subscripts.  If  both  the 
unit  vectors  still  have  subscripts  the  product  is  zero;  if  neither  vector 
has  subscripts  the  product  is  unity,  and  if  one  of  the  vectors  still  has 
subscripts  that  itself  will  be  the  product.  The  sign  is  to  be  taken 
as  that  resulting  from  the  transposition  of  the  subscripts  (see  equa- 
tion (274)),  unless  the  subscript  4  has  been  cancelled,  when  the  sign 
will  be  changed. 

For  example: 

ki24-k34  =  ki2-k3  =  0, 

ki32-k123  =  -  ki23-k123  =  -  1,  (275) 

ki24-k42  =  —  ki24-k24  =  ki. 

It  is  evident  from  these  rules  that  we  may  obtain  the  magnitude 
of  any  desired  component  of  a  vector  by  taking  the  inner  product  of 
the  vector  by  the  corresponding  unit  vector,  it  being  noticed,  of  course, 
that  when  the  unit  vector  involved  contains  the  subscript  4  we  obtain 
the  negative  of  the  desired  component.  For  example,  we  may  obtain 
the  ki2  component  of  a  two-vector  as  follows: 

A12  =  A-ki2  =  (A12k12  -f  Ai3ki8  +  AHku 

(276) 


184.  The  Complement  of  a  Vector.     In  an  n-dimensional  space 
any  m-dimensional  vector  will  uniquely  determine  a  new  vector  of 


Four  Dimensional  Analysis.  199 

dimensions  (n—  m)  which  may  be  called  the  complement  of  the 
original  vector.  The  complement  of  a  vector  may  be  exactly  defined 
as  the  inner  product  of  the  original  vector  with  the  unit  pseudo-scalar 
ki23  ...  „.  In  general,  we  may  denote  the  complement  of  a  vector 
by  placing  an  asterisk  *  after  the  symbol.  As  an  example  we  may 
write  as  the  complement  of  a  two-vector  A  in  our  non-Euclidean  four- 
dimensional  space: 

A*  =  A-ki234  =  (Ai2ki2  +  Ai3k13  +  Ai4ku 

+  AakM  +  A24k24  -f  A.4k,4)  •  k1234    (277) 
Ai3k24  -  Ai4k23  +  A23ku  -f  A24k13  -  A34ki2). 


185.  The  Vector  Operator,  0  or  Quad.     Analogous  to  the  familiar 
three-dimensional  vector-operator  del, 


we  may  define  the  four-dimensional  vector-operator  quad, 

0=k'|r  +  k2i  +  k'^-k<£;-  (279) 

If  we  have  a  scalar  or  a  vector  field  we  may  apply  these  operators 
by  regarding  them  formally  as  one-vectors  and  applying  the  rules 
for  inner  and  outer  multiplication  which  we  have  already  given. 

Thus  if  we  have  a  scalar  function  F  which  varies  continuously 
from  point  to  point  we  can  obtain  a  one-vector  which  we  may  call 
the  four-dimensional  gradient  of  F  at  the  point  in  question  by  simple 
multiplication;  we  have 

dF  dF  dF  3F 

GradF=OF  =  k1-  +  k2-  +  k3--k4-.      (28°) 

If  we  have  a  one-vector  field,  with  a  vector  f  whose  value  varies 
from  point  to  point  we  may  obtain  by  inner  multiplication  a  scalar 
quantity  which  we  may  call  the  four-dimensional  divergence  of  f 
we  have 


Taking  the  outer  product  with  quad  we  may  obtain  a  two-vector,  the 


200  Chapter  Thirteen. 

four-dimensional  curl  of  f, 


By  similar  methods  we  could  apply  quad  to  a  two-vector  function  F 
and  obtain  the  one-vector  function  0  •  F  and  the  three-vector  func- 
tion 0  X  F. 

186.  Still  regarding  0  as  a  one-  vector  we  may  obtain  a  number  of 
important  expressions  containing  0  more  than  once;  we  have: 

0  X  (OF)  =  0,     (283)  0  X  (0  X  f)  =  0,     (286) 

O-(O'F)  =  0,     (284)  0  X  (0  X  F)  =  0,     (287) 

0-(0«r)=0,     (285) 

0-(0  x  f)  =  0(0  -f)  -  (O-O)f,  (288) 

O'(0  X  F)  =  0  X  (0-F)  +  (O'O)F,  (289) 

0-(0  xff)  =  0  x(<0-ff)-(0-0)ff.  (290) 


The  operator  0  '  0  or  O2  nas  l°ng  been  known  under  the  name 
of  the  D'Alembertian, 

62          d2          d2          d2  d2 

(291) 


From  the  definition  of  the  complement  of  a  vector  given  in  the 
previous  section  it  may  be  shown  by  carrying  out  the  proper  expansions 
that 

(0  X0)*  =  0-0*,  (292) 

where  4>  is  a  vector  of  any  number  of  dimensions. 

187.  Tensors.  In  analogy  to  three-dimensional  tensors  we  may 
define  a  four-dimensional  tensor  as  a  quantity  with  sixteen  components 
as  given  in  the  following  table: 

TU     TIZ     TM     jT14, 

TZ\        TZZ        TZS        T'24, 

(293) 

m  rp  m  rp  ^          ' 

1  31        J  32        ^33        1  34, 


^ 


42 


Four  Dimensional  Analysis.  201 

with  the  additional  requirement  that  the  divergence  of  the  tensor, 
defined  as  follows,  shall  itself  be  a  one-vector. 


, 
div  T  =  \  —  —  +  ™  --  h 


dxz         dx3 


dT 


188.  The  Rotation  of  Axes.  Before  proceeding  to  the  application 
of  our  four-dimensional  analysis  to  the  actual  problems  of  relativity 
theory  we  may  finally  consider  the  changes  in  the  components  of  a 
vector  which  would  be  produced  by  a  rotation  of  the  axes.  We  have 
already  pointed  out  that  the  quantity  (xj  +  x22  +  z32  —  £42)  is  an 
invariant  in  our  space  for  any  set  of  rectangular  coordinates  having 
the  same  origin  since  it  is  the  square  of  the  magnitude  of  a  radius 
vector,  and  have  noted  that  in  this  way  we  have  obtained  for  the 
quantity  (z2  +  y2  +  z2  —  cH2)  the  desired  invariance  which  is  charac- 
teristic of  the  Lorentz  transformation.  In  fact  we  may  look  upon 
the  Lorentz  transformation  as  a  rotation  from  a  given  set  of  axes  to  a 
new  set,  with  a  corresponding  re-expression  of  quantities  in  terms  of 
the  new  components.  The  particular  form  of  Lorentz  transformation, 
familiar  in  preceding  chapters,  in  which  the  new  set  of  spatial  axes 
has  a  velocity  component  relative  to  the  original  set,  in  the  .XT-direction 
alone,  will  be  found  to  correspond  to  a  rotation  of  the  axes  in  which 
only  the  directions  of  the  Xi  and  X±  axes  are  changed,  the  Xz  and  X3 
axes  remaining  unchanged  in  direction. 

Let  us  consider  a  one-vector 

a  =  (a*!  +  a2k2  +  a3k3  +  a4ki)  =  (a/k/  +  a2'k2'  +  a/k/  +  a/k/), 

where  a\,  a2,  a3  and  a4  are  the  component  magnitudes,  using  a  set  of 
axes  which  have  ki,  k2,  k3  and  kj  as  unit  vectors  and  a/,  a2r,  a/  and  a/ 
the  corresponding  magnitudes  using  another  set  of  mutually  per- 
pendicular axes  with  the  unit  vectors  k/,  k/,  k/  and  k/.  Our  problem, 


202 


Chapter  Thirteen. 


now,  is  to  find  relations  between  the  magnitudes  ai,  a2,  a3  and  a4  and 
a/,  a2',  0,3  and  a/. 

We  have  already  seen  sections,  (179)  and  (183),  that  we  may  obtain 
any  desired  component  magnitude  of  a  vector  by  taking  its  inner 
product  with  a  unit  vector  in  the  desired  direction,  reversing  the 
sign  if  the  subscript  4  is  involved.  We  may  obtain  in  this  way  an 
expression  for  a\  in  terms  of  a/,  a/,  a/  and  a/.  We  have 

«i  =  a-ki  =  (ai'ki'  +  az'k2'  +  a/k/  +  ai'k4')'ki 

=  ai'ki'  •  kx  +  a2'k2'  •  ki  +  a3'k3'  •  kx  +  a/k/  •  ki.     (295) 

By  similar  multiplications  with  k2,  k3  and  k4  we  may  obtain  expres- 
sions for  a2,  a3,  and  —  a4.  The  results  can  be  tabulated  in  the  con- 
venient form 


a 


kf-k, 


ki'-k 


k/.k, 


a* 


-  ks'-k4 


04 


k4'-k3 


(296) 


Since  the  square  of  the  magnitude  of  the  vector,  (ai2  +  a22  +  a2s 
—  a42),  is  a  quantity  which  is  to  be  independent  of  the  choice  of  axes, 
we  shall  have  certain  relations  holding  between  the  quantities  k/-ki, 
ki'-kj,  etc.  These  relations,  which  are  analogous  to  the  familiar 


Four  Dimensional  Analysis. 


203 


conditions  of  orthogonality  in  Euclidean  space,  can  easily  be  shown 
to  be 


(k/.kj2  +  (k/-k2)2  +  (k/.k,)2  --  (k/-k4)2  =  1, 
(k.'-kO2  +  (k2'-k2)2  +  (k2'-k3)2  -  (k2'.k02  =  1, 
(k/.kO2  +  (k3'-k2)2  +  (k3'-k3)2  -  (k3'-k4)2  =  1, 
(k/.kO2  +  (k/-k2)2  +  (k/'ka)2  -  (k/-k4)2  =  -  1, 


(297) 


and 


+ 


+  (k/.k3)(k2'.k3) 


=0, 


etc.,  for  each  of  the  six  pairs  of  vertical  columns  in  table  (296). 

Since  we  shall  often  be  interested  in  a  simple  rotation  in  which 
the  directions  of  the  X2  and  X3  axes  are  not  changed,  we  shall  be  able 
to  simplify  this  table  for  that  particular  case  by  writing 

k2'  =  k2,         k,'  =  k,, 

and  noting  the  simplifications  thus  introduced  in  the  products  of  the 
unit  vectors,  we  shall  obtain 


•i' 

aj 

a,' 

04' 

al 

k/.fe 

0 

0 

kt'-k, 

a* 

0 

1 

0 

0 

a* 

0 

0 

1 

0 

a4 

-k/-k, 

0 

0 

-  k/-k4 

(298) 


204 


Chapter  Thirteen. 


If  now  we  call  0  the  angle  of  rotation  between  the  two  time  axes 
OX i   and  OX^  we  may  write,  in  accordance  with  equation  (262), 

—  k/-k4  =  cosh  0. 

Since  we  must  preserve  the  orthogonal  relations  (297)  and  may 
also  make  use  of  the  well-known  expression  of  hyperbolic  trigonometry 

cosh2  0  —  sinh2  0  =  1, 
we  may  now  rewrite  our  transformation  table  in  the  form 


a/ 

at' 

a/ 

a/ 

Ol 

cosh  0 

0 

0 

sinh  0 

a2 

0 

1 

0 

0 

a3 

0 

0 

1 

0 

a4 

sinh  0 

0 

0 

cosh  0 

(299) 


By  a  similar  process  we  may  obtain  transformation  tables  for  the 
components  of  a  two-vector  A.  Expressing  A  in  terms  of  the  unit 
vectors  ki2',  W,  ki/,  etc.,  and  taking  successive  inner  products  with 
the  unit  vectors  ki2,  ki3,  ki4,  etc.,  we  may  obtain  transformation 
equations  which  can  be  expressed  by  the  tabulation  (300)  shown  on 
the  following  page. 


Four  Dimensional  Analysis. 


205 


V 

., 

, 

A  23' 

,„- 

AM' 

A. 

w*. 

k,j'-k12 

k14'-k12 

k23'-k12 

r* 

V.,,, 

Kl2    *  *£l3 

k»'-kB 

w* 

w* 

w,. 

AM 

-W-kn 

-k14'-k14 

-w,,. 

-w* 

-k3/-k14 

^23 

>.,.„ 

ku'-ka 

,,,, 

w* 

w-k. 

k34'-k23 

A24 

-k12'-k24 

-k13'-k24 

-,,,„ 

-k,,_ 

-y* 

-k34/-k24 

*, 

-w* 

-,,•,„ 

-w* 

-*•* 

Tw* 

-k?4'-k34 

(300) 


For  the  particular  case  of  a  rotation  in  which  the  direction  of  the 
Xz  and  X3  axes  are  not  changed  we  shall  have 

k2   =  k2,         k3   =  k3, 

and  very  considerable  simplification  will  be  introduced.     We  shall 
have,  for  example, 

ku'-ku  =  (k/  x  k2')-(k!  X  k2)  =  (k/  X  k2).(k!  X  k2)  =  k/-^, 
k13'-k12  =  (k/  X  ks')-(ki  X  k2)  =  (k/  X  k3).(k!  X  k2)  =  0, 
etc. 
Making  these  and  similar  substitutions  and  introducing,  as  before, 


206 


Chapter  Thirteen. 


the  relation  —  k'4-k4  =  cosh  0  where  <f>  is  the  non-Euclidean  angle 
between  the  two  time  axes,  we  may  write  our  transformation  table 
in  the  form 


A»> 

A  13 

A, 

y 

A24' 

£ 

A  12 

cosh  0 

0 

0 

0 

sinh  0 

0 

A» 

0 

cosh  0 

0 

0 

0 

sinh  0 

AU 

0 

0 

1 

0 

0 

0 

An 

0 

0 

0 

1 

0 

0 

A24 

—  sinh  0 

0 

0 

0 

cosh  0 

0 

A, 

0 

—  sinh  0 

0 

0 

0 

cosh  0 

(301) 


189.  Interpretation  of  the  Lorentz  Transformation  as  a  Rotation 
of  Axes.  We  may  now  show  that  the  Lorentz  transformation  may 
be  looked  upon  as  a  change  from  a  given  set  of  axes  to  a  rotated  set. 

Since  the  angle  0  which  occurs  in  our  transformation  tables  is 
that  between  the  k4  axis  and  the  new  k/  axis,  we  may  write,  in  ac- 
cordance with  equations  (265)  and  (266), 


cosh  0  =  — 


r— !•• 

V1-  •? 


sinh  0 


•1    - 

• 


where  V  is  the  velocity  between  the  two  sets  of  space  axes  which 
correspond  to  the  original  and  the  rotated  set  of  four-dimensional 
axes.  This  will  permit  us  to  rewrite  our  transformation  table  for  the 


Four  Dimensional  Analysis. 
components  of  a  one-vector  in  the  forms 


a* 


ai 


1 


Vc 


a2 


a 


fll 


a 
a3' 


at 


fll 
1 


/       F2 

1±I 

0 
0 

~wr 


a2 


a, 


V/c 


a* 


-F/c 


72 


207 


(302) 


Consider  now  any  point  P(#i,  £2,  #3,  £4).  The  radius  vector  from 
the  origin  to  this  point  will  be  r  =  (ziki  +  x2k2  +  ^k3  +  x4k4),  or, 
making  use  of  the  relations  between  rci,  xz,  xs,  x±  and  x,  y,  z,  t  given 
by  equations  (252),  we  may  write 

r  =  (zk,  +  i/k2  +  2k3  +  d^). 

Applying  our  transformation  table  to  the  components  of  this  one- 
vector,  we  obtain  the  familiar  equations  for  the  Lorentz  transformation 

x  -  Vt 


208  Chapter  Thirteen. 


y  =  y, 

yf  =  y 
j/   


i      /      v   \ 

^If-?*) 


We  thus  see  that  the  Lorentz  transformation  is  to  be  interpreted 
in  our  four-dimensional  analysis  as  a  rotation  of  axes. 

190.  Graphical  Representation.  Although  we  have  purposely  re- 
stricted ourselves  in  the  foregoing  treatment  to  methods  of  attack 
which  are  almost  purely  analytical  rather  than  geometrical  in  nature, 
the  importance  of  a  graphical  representation  of  our  four-dimensional 
manifold  should  not  be  neglected.  The  difficulty  of  representing  all 
four  axes  on  a  single  piece  of  two-dimensional  paper  is  not  essentially 
different  from  that  encountered  in  the  graphical  representation  of  the 
facts  of  ordinary  three-dimensional  solid  geometry,  and  these  diffi- 
culties can  often  be  solved  by  considering  only  one  pair  of  axes  at  a 
time,  say  OXi  and  OX*,  and  plotting  the  occurrences  in  the  XiOX^ 
plane.  The  fact  that  the  geometry  of  this  plane  is  a  non-Euclidean 
one  presents  a  more  serious  complication  since  the  figures  that  we 
draw  on  our  sheet  of  paper  will  obviously  be  Euclidean  in  nature, 
but  this  difficulty  also  can  be  met  if  we  make  certain  conventions  as 
to  the  significance  of  the  lines  we  draw,  conventions  which  are  funda- 
mentally not  so  very  unlike  the  conventions  by  which  we  interpret  as 
solid,  a  figure  drawn  in  ordinary  perspective.  /^ 

Consider  for  example  the  diagram  shown  in  figure  18,  where  we 
have  drawn  a  pair  of  perpendicular  axes,  OX1}  and  OX*,  and  the 
two  unit  hyperbolae  given  by  the  equations 

x?  -  x?  =  1, 

(303) 

Xf  -  X?   =    -   1, 

together  with  their  asymptotes,  OA  and  OB,  given  by  the  equation 

xf  -  x?  =  0.  (304) 

This  purely  Euclidean  figure  permits,  as  a  matter  of  fact,  a  fairly 
satisfactory  representation  of  the  non-Euclidean  properties  of  the 
manifold  with  which  we  have  been  dealing. 


Four  Dimensional  Analysis. 


209 


OX  i  and  OX4  may  be  considered  as  perpendicular  axes  in  the 
non-Euclidean  XiOX\  plane.  Radius  vectors  lying  in  the  quadrant 
A  OB,  will  have  a* greater  component  along  the  A%  than  along  the  X\ 
axis  and  hence  will  be  5-vectors  with  the  magnitude  s  =  Vz42  —  zr, 
where  xi  and  x4  are  the  coordinates  of  the  terminal  of  the  vector. 


FIG.  18. 

7-radius-vectors  will  lie  in  the  quadrant  BOG  and  will  have  the  mag- 
nitude s  =  Vxi2  —  z42.  Radius  vectors  lying  along  the  asymptotes 
OA  and  OB  will  have  zero  magnitudes  (s  =  V#i2  —  x£  =  0)  and 
hence  will  be  singular  vectors. 

Since  the  two  hyperbolae  have  the  equations  x-c  —  x<?  =  1  and 
Xi~  —  x?  =  --  1,  rays  such  as  Oa,  Oa',  Ob,  etc.,  starting  from  the 
origin  and  terminating  on  the  hyperbolae,  will  all  have  unit  magnitude. 
Hence  we  may  consider  the  hyperbolae  as  representing  unit  pseudo- 
circles  in  our  non-Euclidean  plane  and  consider  the  rays  as  repre- 
senting the  radii  of  these  pseudo-circles. 

A  non-Euclidean  rotation  of  axes  will  then  be  represented  by 
changing  from  the  axes  OXi  and  OX*  to  OXi  and  OXi,  and  taking 
Oa!  and  Ob'  as  unit  distances  along  the  axes  instead  of  Oa  and  Ob. 

15 


210  Chapter  Thirteen. 

It  is  easy  to  show,  as  a  matter  of  fact,  that  such  a  change  of  axes 
and  units  does  correspond  to  the  Lorentz  transformation.  Let  x\ 
and  x4  be  the  coordinates  of  any  point  with  respect  to  the  original 
axes  OXi  and  OX4,  and  x\'  and  x4"  the  coordinates  of  the  same  point 
referred  to  the  oblique  axes  OXi  and  OX4f,  no  change  having  yet 
been  made  in  the  actual  lengths  of  the  units  of  measurement.  Then, 
by  familiar  equations  of  analytical  geometry,  we  shall  have 

XL  =  Xi"  cos  6  +  x4"  sin  0, 

(305) 
X4  =  XL"  sin  0  +  x"  cos  0, 

where  0  is  the  angle  XiOXi. 

We  have,  moreover,  from  the  properties  of  the  hyperbola, 

0o_'  _  Ob^  _  J_ 

Oa  ~  Ob  ~~ 


and  hence  if  we  represent  by  x\  and  x±  the  coordinates  of  the  point 
with  respect  to  the  oblique  axes  and  use  Oa'  and  Ob'  as  unit  distances 
instead  of  Oa  and  Ob,  we  shall  obtain 

cos  0  sin  0 

==  + 


Vcos2  0  -  sin2  0  Vcos2  0  -  sin2  0 ' 

,  sin  0  7  cos  0 

Vcos2  0  —  sin2  0  Vcos2  0  —  sin2  0 ' 

It  is  evident,  however,  that  we  may  write 

sin  0  dxi      V 

-  =  tan  0  =  -T- -  =  - , 
cos  0  az4       c 

where  V  may  be  regarded  as  the  relative  velocity  of  our  two  sets  of 
space  axes.  Introducing  this  into  the  above  equations  and  also 
writing  x\  —  x,  x4  =  ctt  x\  =  x',  x4  =  ctf,  we  may  obtain  the  familiar 
equations 


l~^ 


Four  Dimensional  Analysis.  211 

We  thus  see  that  our  diagrammatic  representation  of  non-Euclidean 
rotation  in  the  XiOX*  plane  does  as  a  matter  of  fact  correspond  to 
the  Lorentz  transformation. 

Diagrams  of  this  kind  can  now  be  used  to  study  various  kine- 
matical  events.  5-curves  can  be  drawn  in  the  quadrant  AOB  to  repre- 
sent the  space-time  trajectories  of  particles,  their  form  can  be  in- 
vestigated using  different  sets  of  rotated  axes,  and  the  equations  for 
the  transformation  of  velocities  and  accelerations  thus  studied. 
7-lines  perpendicular  to  the  particular  time  axis  used  can  be  drawn  to 
correspond  to  the  instantaneous  positions  of  actual  lines  in  ordinary 
space  and  studies  made  of  the  Lorentz  shortening.  Singular  vectors 
along  the  asymptote  OB  can  be  used  to  represent  the  trajectory  of  a 
ray  of  light  and  it  can  be  shown  that  our  rotation  of  axes  is  so  devised 
as  to  leave  unaltered,  the  angle  between  such  singular  vectors  and  the 
0X4  axis,  corresponding  to  the  fact  that  the  velocity  of  light  must 
appear  the  same  to  all  observers.  Further  development  of  the  possi- 
bilities of  graphical  representation  of  the  properties  of  our  non- 
Euclidean  space  may  be  left  to  the  reader. 

PART  II.     APPLICATIONS  OF  THE  FOUK-DIMENSIONAL  ANALYSIS. 

191.  We   may   now   apply   our   four-dimensional   methods   to    a 
number  of  problems  in  the  fields  of  kinematics,  mechanics  and  electro- 
magnetics.    Our  general  plan  will  be  to  express  the  laws  of  the  par- 
ticular field  in  question  in  four-dimensional  language,  making  use  of 
four-dimensional  vector  quantities  of  a  kinematical,  mechanical,  or 
electromagnetic    nature.     Since    the    components    of    these    vectors 
along  the  three  spatial  axes  and  the  temporal  axis  will  be  closely 
related  to  the  ordinary  quantities  familiar  in  kinematical,  mechanical, 
and  electrical  discussions,  there  will  always  be  an  easy  transition  from 
our  four-dimensional  language  to  that  ordinarily  used  in  such  dis- 
cussions, and  necessarily  used  when  actual  numerical  computations 
are  to  be  made.     We  shall  find,  however,  that  our  four-dimensional 
language  introduces  an  extraordinary  brevity  into  the  statement  of  a 
number  of  important  laws  of  physics. 

KINEMATICS. 

192.  Extended  Position.     The  position  of  a  particle  and  the  par- 
ticular instant  at  which  it  occupies  that  position  can  both  be  indi- 


212  Chapter  Thirteen. 

cated  by  a  point  in  our  four-dimensional  space.  We  can  call  this 
the  extended  position  of  the  particle  and  determine  it  by  stating  the 
value  of  a  four-dimensional  radius  vector 

r  =  (siki  +  z2k2  +  z3k3  +  z4k4).  (306) 

193.  Extended  Velocity.  Since  the  velocity  of  a  real  particle  can 
never  exceed  that  of  light,  its  changing  position  in  space  and  time 
will  be  represented  by  a  5-curve. 

The  equation  for  a  unit  vector  tangent  to  this  5-curve  will  be 


where  ds  indicates  interval  along  the  5-curve;  and  this  important 
vector  w  may  be  called  the  extended  velocity  of  the  particle. 
Remembering  that  for  a  5-curve 


ds  =     dz42  -  dxS  -  dx<?  -  dxj  =  cdt  A/1  -  -,          (308) 

.          c 

we  may  rewrite  our  expression  for  extended  velocity  in  the  form 

(309) 


where  u  is  evidently  the  ordinary  three-dimensional  velocity  of  the 
particle. 

Since  w  is  a  four-dimensional  vector  in  our  imaginary  space,  we 
may  use  our  tables  for  transforming  the  components  of  w  from  one 
set  of  axes  to  another.  We  shall  find  that  we  may  thus  obtain  trans- 
formation equations  for  velocity  identical  with  those  already  familiar 
in  Chapter  IV. 

The  four  components  of  w  are 


k4 


and  with  the  help  of  table  (302)  we  may  easily  obtain,  by  making 
simple  algebraic  substitutions,  the  following  familiar  transformation 


Four  Dimensional  Analysis. 


213 


equations : 


,       ux-V 

uxf  = 

1  - 


1  - 


uxV 


I  _  UxV 


i  - 


uzV 


This  is  a  good  example  of  the  ease  with  which  we  can  derive  our 
familiar  transformation  equations  with  the  help  of  the  four-dimensional 
method. 

194.  Extended  Acceleration.  We  may  define  the  extended  accel- 
eration of  a  particle  as  the  rate  of  curvature  of  the  6-line  which  deter- 
mines its  four-dimensional  position.  We  have 


d?i      dw      d 
ds2=~ds=Ts 


Or,  introducing  as  before  the  relation  ds  =  cdl  A/  1  -    — ,  we  may  write 


(310) 


du 

j±    i 


1          u  du 


u*\dt 


+ 


u  du 


uz\ 

-  <*) 


(311) 


214  Chapter  Thirteen. 

where  u  is  evidently  the  ordinary  three-dimensional  velocity,  and  -r- 

CtL 

the  three-dimensional  acceleration;  and  we  might  now  use  our  trans- 
formation table  to  determine  the  transformation  equations  for  accel- 
eration which  we  originally  obtained  in  Chapter  IV. 

195.  The  Velocity  of  Light.  As  an  interesting  illustration  of  the 
application  to  kinematics  of  our  four-dimensional  methods,  we  may 
point  out  that  the  trajectory  of  a  ray  of  light  will  be  represented  by  a 
singular  line.  Since  the  magnitude  of  all  singular  vectors  is  zero  by 
definition,  we  have  for  any  singular  line 

dx^  +  dxt*  +  dx^  =  dxf, 

or,  since  the  magnitude  will  be  independent  of  any  particular  choice 
of  axes,  we  may  also  write 

dxi'*  +  dx2/2  +  dx3'2  =  dxf, 
Transforming  the  first  of  these  equations  we  may  write 

dxf  +  dx22  +  dx32  _  dx2  +  dy2  +  dz*  _ 

dx?  c*dt2 

or 

dl 


Similarly  we  could  obtain  from  the  second  equation 

dl' 

dt  =  C' 

We  thus  see  that  a  singular  line  does  as  a  matter  of  fact  correspond 
to  the  four-dimensional  trajectory  of  a  ray  of  light  having  the  velocity  c, 
and  that  our  four-dimensional  analysis  corresponds  to  the  require- 
ments of  the  second  postulate  of  relativity  that  a  ray  of  light  shall 
have  the  same  velocity  for  all  reference  systems. 

THE  DYNAMICS  OF  A  PARTICLE. 

196.  Extended  Momentum.  We  may  define  the  extended  momen- 
tum of  a  material  particle  as  equal  to  the  product  m0w  of  its  mass  m0, 
measured  when  at  rest,  and  its  extended  velocity  w.  In  accordance 


Four  Dimensional  Analysis.  215 

with  equation  (309)  for  extended  velocity,  we  may  write  then,  for 
the  extended  momentum, 

(312) 


1-? 

Or,  if  in  accordance  with  our  considerations  of  Chapter  VI  we  put 
for  the  mass  of  the  particle  at  the  velocity  u 


••-? 


we  may  write 


m  = 


m0w  =  m-  +  mkt.  (313) 


We  note  that  the  space  component  of  this  vector  is  ordinary  momen- 
tum and  the  time  component  has  the  magnitude  of  mass,  and  by 
applying  our  transformation  table  (302)  we  can  derive  very  simply 
the  transformation  equations  for  mass  and  momentum  already 
obtained  in  Chapter  VI. 

197.  The  Conservation  Laws.  We  may  now  express  the  laws  for 
the  dynamics  of  a  system  of  particles  in  a  very  simple  form  by  stating 
the  principle  that  the  extended  momentum  of  a  system  of  particles  is  a 
quantity  which  remains  constant  in  all  interactions  of  the  particles, 
we  have  then 


2ra0w  =  2  (      -  +  wk4  J  =  a  constant, 


(314) 


where  the  summation  2  extends  over  all  the  particles  of  the  system. 
It  is  evident  that  this  one  principle  really  includes  the  three 
principles  of  the  conservation  of  momentum,  mass,  and  energy. 
This  is  true  because  in  order  for  the  vector  2ra0w  to  be  a  constant 
quantity,  its  components  along  each  of  the  four  axes  must  be  con- 
stant, and  as  will  be  seen  from  the  above  equation  this  necessitates 
the  constancy  of  the  momentum  2wu,  of  the  total  mass  2w,  and  of 

the  total  energy  2  —  . 


216  Chapter  Thirteen. 

THE  DYNAMICS  OF  AN  ELASTIC  BODY. 

Our  four-dimensional  methods  may  also  be  used  to  present  the 
results  of  our  theory  of  elasticity  in  a  very  compact  form. 

198.  The  Tensor  of  Extended  Stress.     In  order  to  do  this  we  shall 
first  need  to  define  an  expression  which  may  be  called  the  four-dimen- 
sional stress  in  the  elastic  medium.     For  this  purpose  we  may  take  the 
symmetrical  tensor  Tm  defined  by  the  following  table: 

PXX       Pxy       PXZ       CgX) 
Pyx      Jpyy      Pvz      ^Qyj 

Pzx     Pzy     Pzz     cgz,  (315) 

w, 

1C  C  C 

where  the  spatial  components  of  Tm  are  equal  to  the  components  of 
the  symmetrical  tensor  p  which  we  have  already  defined  in  Chapter 
X  and  the  time  components  are  related  to  the  density  of  momentum  g, 
density  of  energy  flow  s  and  energy  density  w,  as  shown  in  the  tabu- 
lation. 

From  the  symmetry  of  this  tensor  we  may  infer  at  once  the  simple 
relation  between  density  of  momentum  and  density  of  energy  flow: 

g  =  | ,  (316) 

with  which  we  have  already  become  familiar  in  Section  132. 

199.  The  Equation  of  Motion.     We  may,  moreover,   express  the 
equation  of  motion  for  an  elastic  medium  unacted  on  by  external 
forces  in  the  very  simple  form 

div  Tm  =  0.  (317) 

It  will  be  seen  from  our  definition  of  the  divergence  of  a  four- 
dimensional  tensor,  Section  187,  that  this  one  equation  is  in  reality 
equivalent  to  the  two  equations 

div  p  +  ~  =  0  (318) 

and 

dw 
div  s  +  —  =  0. 

ot 


Four  Dimensional  Analysis.  217 

The  first  of  these  equations  is  identical  with  (184)  of  Chapter  X, 
which  we  found  to  be  the  equation  for  the  motion  of  an  elastic  medium 
in  the  absence  of  external  forces,  and  the  second  of  these  equations 
expresses  the  principle  of  the  conservation  of  energy. 

The  elegance  and  simplicity  of  this  four-dimensional  method  of 
expressing  the  results  of  our  laborious  calculations  in  Chapter  X  can- 
not fail  to  be  appreciated. 

ELECTROMAGNETICS. 

We  also  find  it  possible  to  express  the  laws  of  the  electromagnetic 
field  very  simply  in  our  four-dimensional  language. 

200.  Extended  Current.  We  may  first  define  the  extended  current, 
a  simple  but  important  one-vector,  whose  value  at  any  point  will  de- 
pend on  the  density  and  velocity  of  charge  at  that  point.  We  shall 
take  as  the  equation  of  definition 

q  =  Pow  =  P  I"  +  k4  I  ,  (319) 

where 


is  the  density  of  charge  at  the  point  in  question. 

201.  The  Electromagnetic  Vector  M.  We  may  further  define  a 
two-vector  M  which  will  be  directly  related  to  the  familiar  vectors 
strength  of  electric  field  e  and  strength  of  magnetic  field  h  by  the 
equation  of  definition 

M  =  (hikw  +  ft2k3i  +  ^3^12  -  eiki4  -  e2k24  -  e3k34) 

or  (320) 

M*  =  (eikaa  +  e2k3].  +  ^ki2  +  Aik14  +  /*2k24  +  /*3k34), 


where  e\,  e«,  e3,  and  hi,  hz,  h3  are  the  components  of  e  and  h. 

202.  The  Field  Equations.     We  may  now  state  the  laws  of  the 
electromagnetic  field  in  the  extremely  simple  form 

0-M  =  q,  (321) 

0  X  M  =  0.  (322) 


218  Chapter  Thirteen. 

These  two  simple  equations  are,  as  a  matter  of  fact,  completely 
equivalent  to  the  four  field  equations  which  we  made  fundamental 
for  our  treatment  of  electromagnetic  theory  in  Chapter  XII.  Indeed 
if  we  treat  <0  formally  as  a  one-vector 

and  apply  it  to  the  electromagnetic  vector  M  expressed  in  the  extended 
form  given  in  the  equation  of  definition  (320)  we  shall  obtain  from 
(321)  the  two  equations 

19e         u 
curl  h  —  -  —  =  p  -  , 

div  e  =  p, 
and  from  (322) 

div  h  =  0, 

1  dh 

curle  +  --  =  0, 

where  we  have  made  the  substitution  #4  =  ct.  These  are  of  course 
the  familiar  field  equations  for  the  Maxwell-Lorentz  theory  of  electro- 
magnetism. 

203.  The  Conservation  of  Electricity.     We  may  also  obtain  very 
easily  an  equation  for  the  conservation  of  electric  charge.     In  accord- 
ance with  equation  (284)  we  may  write  as  a  necessary  mathematical 
identity 

O-(O'M)  =  0.  (323) 

Noting  that  <0  •  M  =  q,  this  may  be  expanded  to  give  us  the  equation 
of  continuity. 

div  pu  +  j£  =  0.  (324) 

204.  The  Product  M  •  q.     We  have  thus  shown  the  form  taken  by 
the  four  field  equations  when  they  are  expressed  in  four  dimensional 
language.     Let  us  now  consider  with  the  help  of  our  four-dimensional 
methods  what  can  be  said  about  the  forces  which  determine  the 
motion  of  electricity  under  the  action  of  the  electromagnetic  field. 

Consider  the  inner  product  of  the  electromagnetic  vector  and 


Four  Dimensional  Analysis.  219 

the  extended  current: 


f 
=  p( 


-+  k4 

(325) 

[uXh]*l         e-u. 
e+-—  -j  +  p—  k, 


We  see  that  the  space  component  of  this  vector  is  equal  to  the  ex- 
pression which  we  have  already  found  in  Chapter  XII  as  the  force 
acting  on  the  charge  contained  in  unit  volume,  and  the  time  com- 
ponent is  proportional  to  the  work  done  by  this  force  on  the  moving 
charge;  hence  we  may  write  the  equation 

(326) 

an  expression  which  contains  the  same  information  as  that  given  by 
the  so-called  fifth  fundamental  equation  of  electromagnetic  theory, 
f  being  the  force  exerted  by  the  electromagnetic  field  per  unit  volume 
of  charged  material. 

205.  The  Extended  Tensor  of  Electromagnetic  Stress.  We  may 
now  show  the  possibility  of  defining  a  four-dimensional  tensor  Te  such 
that  the  important  quantity  M-q  shall  be  equal  to  —  div  Te.  This 
will  be  valuable  since  we  shall  then  be  able  to  express  the  equation 
of  motion  for  a  combined  mechanical  and  electrical  system  in  a.  very 
simple  and  beautiful  form. 

Consider  the  symmetrical  tensor 


JTI  rji  rri 

I"    I*    !>  (327) 

J-  32        J-  33         J-  34) 


defined  by  the  expression 

Tik  =  \{MnMkl  +  MiZMk2  +  MflMkz  -  M^Mk, 

(328) 
+  M}l*Mkl*  +  M,-2*MM*  +  Af,,*Aftt* 

where  j,  k  =  1,  2,  3,  4. 


220  Chapter  Thirteen. 

It  can  then  readily  be  shown  by  expansion  that 

-  divT.  =  M-(O-M)  +M*-(0-M*). 

But,  in  accordance  with  equations  (321),  (326),  (292)  and  (322),  this 
is  equivalent  to 


-divTe  =  M-q  =  {f +  -^k4}. 


(329) 


Since  in  free  space  the  value  of  the  force  f  is  zero,  we  may  write 
for  free  space  the  equation 

div  T.  =  0.  (330) 

This  one  equation  is  equivalent,  as  a  matter  of  fact,  to  two  im- 
portant and  well-known  equations  of  electromagnetic  theory.  If  we 
develop  the  components  TU,  TIZ,  etc.,  of  our  tensor  in  accordance 
with  equations  (328)  and  (320)  we  find  that  we  can  write 


Te 


txx        txy 

**, 

So, 

c  ' 

tyX      tyy 

+» 

f» 

c  J 

S2 

tzX        tzy 

*" 

c  J 

Sx                Sy 

if 



ty, 

.   C            C 

c 

(331) 


where  we  shall  have 


etc. 

(332) 
Sx  =  c(eyhx  —  ezhy), 

etc. 

w  —  i(g2  ~h  ^2)> 

4/  thus  being  equivalent  to  the  well-known  Maxwell  three-dimensional 
stress  tensor,  szf  sy,  etc.,  being  the  components  of  the  Poynting  vector 
c[e  X  h]*,  and  w  being  the  familiar  expression  for  density  of  electro- 


Four  Dimensional  Analysis.  221 


magnetic  energy  —  -  —  .     We  thus  see  that  equation  (330)  is  equiva- 

• 

lent  to  the  two  equations 

1  ds 


dw 

div  s  +  —  =  0. 
ot 

The  first  of  these  is  the  so-called  equation  of  electromagnetic  momen- 
tum, and  the  second,  Poynting's  equation  for  the  flow  of  electromag- 
netic energy. 

206.  Combined  Electrical  and  Mechanical  Systems.  For  a  point 
not  in  free  space  where  mechanical  and  electrical  systems  are  both 
involved,  taking  into  account  our  previous  considerations,  we  may 
now  write  the  equation  of  motion  for  a  combined  electrical  and 
mechanical  system  in  the  very  simple  form 

div  Tm  +  div  Te  =  0. 

And  we  may  point  out  in  closing  that  we  may  reasonably  expect  all 
forces  to  be  of  such  a  nature  that  our  most  general  equation  of  motion 
for  any  continuous  system  can  be  written  in  the  form 

div  Ti  +  div  T2  +  •  •  •  =  0. 


APPENDIX  I. — SYMBOLS  FOR  QUANTITIES. 

Scalar  Quantities.     (Indicated  by  Italic  type.) 

c  speed  of  light. 

e  electric  charge. 

E  energy. 

H  kinetic  potential. 

K  kinetic  energy. 

I,  m,  n  direction  cosines. 

L  Lagrangian  function. 

p  pressure. 

Q  quantity  of  electricity. 

S  entropy. 

t  time. 


T  temperature,  function  2m0c2  (  1  —  \/ 1 J 

U  potential  energy. 

v  volume. 

V  relative  speed  of  coordinate  systems,  volume. 

w  energy  density. 

W  work. 

€  dielectric  constant. 
1 


/        .V2' 

V1-  -j 


V  index  of  refraction,  magnetic  permeability. 

v  frequency. 

p  density  of  charge. 

cr  electrical  conductivity. 

0  non-Euclidean  angle  between  time  axes. 

010203-  •  •  generalized  coordinates. 

\l/  scalar  potential. 

\l/i\l/2\l/3-  •  •  generalized  momenta. 

222 


Appendix  I.  223 


Vector  Quantities.     (Indicated  by  Clarendon  type.) 

B  magnetic  induction. 

c  extended  acceleration. 

D  dielectric  displacement, 

e  electric  field  strength  in  free  space. 

E  electric  field  strength  in  a  medium, 

f  force  per  unit  volume. 

F  force  acting  on  a  particle, 

g  density  of  momentum. 

h  magnetic  field  strength  in  free  space. 

H  magnetic  field  strength  in  a  medium. 

i  density  of  electric  current. 

M  angular  momentum,  electromagnetic  vector. 

p  symmetrical  elastic  stress  tensor, 

q  extended  current, 

r  radius  vector 

s  density  of  energy  flow, 

t  unsymmetrical  elastic  stress  tensor, 

u  velocity, 

w  extended  velocity. 

<t>  vector  potential. 


APPENDIX   II.— VECTOR  NOTATION. 

Three  Dimensional  Space. 
Unit  Vectors,  i  j  k 
Radius  Vector,  r  =  xi  +  yj  +  zk 
Velocity, 

dr 
u  =  jj  =  xi  +  yj  -f-  zk 

Acceleration, 

=     WX1     -f-    Uy]     -\~    UZK. 

Inner  Product, 

a-b  =  axbx  +  a^^  +  azbz 
Outer  Product, 

a  X  b  =  (axby  —  aybx)ij  +  (aj/6^  —  azby)jk  +  (a26x 
Complement  of  Outer  Product, 
[a  X  b]*  =  (aybz  -  azby)i  +  (azbx  -  axbz)j  +  (a^ 
The  Vector  Operator  Del  or  V, 


.dA       .dA      ,  dA 
gradA=  VA=i-  +  J-+k- 


(  daz      day\.       (dax      daz\  . 
curia  =  [V  X  a]*  =  (-  -—  Ji+  (~  —-  J  j 


224 


Appendix  II.  225 


Non-Euclidean  Four  Dimensional  Space. 

Unit  Vectors,  ki  k2  ks  k4 
Radius  Vector, 

r  =  ziki  +  z2k2 

=  xi  +  yj  +  zk  +  ctk, 
One  Vector, 

a  =  aikj  +  a2k2  +  a3k3  + 
Two  Vector, 

A  =  Ai2ki2  +  Ai3ki3  +  Ai4ki4  +  A23k2s 
Three  Vector, 

H  =  Hi23k123  +  5Ii24k124  +  5I134k134 

Pseudo  Scalar, 

a  =  aki234 

Transposition  of  Subscripts, 

...  =  —  Iw...  =  k6ca 


Inner  Product  of  One  Vectors, 

(See  Section  183). 
Outer  Product  of  One  Vectors, 

^a6«*«    /N   K-nm'" 

Complement  of  a  Vector, 

0*  =  <£ 
The  Vector  Operator  Quad  or  0, 


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